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DUREN, Lowell Reid, 1940- AN ADAPTATION OF THE MOORE METHOD TO THE TEACHING OF UNDERGRADUATE REAL ANALYSIS— A CASE STUDY REPORT.

The Ohio State University, Ph.D., 1970 Education, theory and practice

University Microfilms, A XERQXCompany, Ann Arbor, Michigan

© Copyright by

Lowe 11 Re i d Du ren

1970

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED AN ADAPTATION OF THE MOORE METHOD TO THE TEACHING

OF UNDERGRADUATE REAL ANALYSIS—

A CASE STUDY REPORT

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Lowe 11 Reid Duren, B.S., M.N.S.

******

The Ohio State University 1970

Approved by

Adviser College of Education ACKNOWLEDGMENTS

I wish to express my thanks to my adviser, Dr. Harold Trimbl

Successful graduate study at Ohio State would have been impossible for me without his patience and encouragement. The completion of this dissertation owes much to Dr. Trimble's assistance and confi­ dence in me.

Thanks are also due to the other members of my reading committee, Dr. Herbert Coon and Dr. Alan Osborne. In a large, some­ times impersonal, university, it is reassuring to find such warm and human gentlemen.

I am indeed grateful to my colleagues at Western Maryland college, Professors Spicer, Sorkin, Lightner, Jordy, and Eshleman, for their conversations and assistance in gathering information relevant to this study, and their moral support during completion of it . I especially want to thank my departmental chairman, Dr.

James Lightner, for his many valuable comments, assistance, and encouragement.

Special recognition is also due the students: Sue Robertson

Randy Klinger, Robert Gagnon, Kevin Fried, Bill E llio tt, Owen Ecker,

Ray Brown, and Dave Baugh.

Finally, I wish to thank my wife, Reba, for putting up with me through years of study and the completion of this dissertation. VITA

September 22, 19A O ...... Born, Wynnewood, Oklahoma

1962 ...... B.S., Southwestern State College Weatherford, Oklahoma

1962-1963 ...... Teacher, Sayre Junior College Sayre, Oklahoma

1963-1965 ...... Teacher, Moore High School Moore, Oklahoma

1965 ...... M.N.S., University of Oklahoma Norman, Oklahoma

1965-1966 ...... National Science Foundation Academic Year Institute, The Ohio State University, Columbus, Ohio

1966-1968 ...... Assistant Professor, Department of Mathematics, Wisconsin State University, Whitewater, Wisconsin

1968- ...... Assistant Professor, Department of Mathematics, Western Maryland College, Westminster, Maryland

FIELDS OF STUDY

Major Field: Mathematics Education

Studies in Mathematics Education. Professors Harold C. Trimble and Alan R. Osborne

Studies in Teacher Education. Professor Herbert L. Coon

i i i TABLE OF CONTENTS

ACKNOWLEDGMENTS

VITA

LIST OF TABLES v

LIST OF ILLUSTRATIONS vi

Chapter INTRODUCTION

Defini tions More on Moore Some Questions Origin and Background of the Problem Statement of the Problem Review of Related Literature Design of the Study The course The students Data collected Course content Organization of the Dissertation

11 THE COURSE 23

Progress Through the Course Introductory material Section five Section six Picture "proofs" The remaining material The hour test Daily progress Conduct of the Classes Reactions of the Instructor The classes The interviews Summary

iv I I I . STUDENT PERFORMANCE *f3

Student Number One Backg round Performance Student Number Two Background Performance Student Number Three Background Performance Student Number Four Background Performance Student Number Five Background Performance Student Number Six Background Performance Student Number Seven Background Performance Student Number Eight Background Performance Summary

IV. EXAMINATION OF QUESTIONS...... 78

The Questions Summary

V. INFORMAL CONCLUSIONS ...... 85

Limitations and Disclaimers Data sources Influence of grades Influence of the study Influence of friendship The informal nature Specificity of the class Comparison with lecture Some Conclusions Opinions and Questions The role of the instructor The personality of the instructor Proof techniques Independent study and learning Concluding Remarks

v APPENDIX

A...... * ...... 100

B...... 118

C...... 125

D...... 129

E...... 135

F...... 144

G. , ...... 156

BIBLIOGRAPHY ...... 19*

v I LIST OF TABLES

Table Page 1. Class P r o f ile ...... 18

2. Dai ly P ro g re s s...... 33

3. Work and Achievement ...... ^6

vi i LIST OF ILLUSTRATIONS

Figure Page 1. Picture Proof ...... 29

2. An Exception ...... 30

vi i i CHAPTER I

INTRODUCTION

It is commonly accepted that for a student to learn mathe­ matics he must do mathematics. The ways in which students are led to do mathematics are quite varied and there is no unique method.

In fact, the methods used to teach collegiate mathematics are prob­ ably as diverse as the subject matter its e lf and the people teaching it. This is not necessarily bad, for as Polya states: ". . . there are as many good methods as there are good teachers." [35: 114].^

Still it seems that mathematics teachers, in spite of their different teaching methods, have a common goal. That is, they want to get the students actively involved in doing mathematics. Moise states this well when he says:

We can "cover" very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has "studied" your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does. [29: *»09l •

Teaching can be defined as a dialogue between the teacher and the student. By considering forms this dialogue takes, one is able to examine how the student is led to do mathematics. For exam­ ple, two popular methods of teaching collegiate mathematics are the

Whe firs t set of digits represents the number of the biblio­ graphic entry; the second represents the page number. "lecture method" and the "Moore method."' The choice of these two methods as examples was not motivated by a desire to show one method superior to the other, and no such claim will be made in this disser­ tation. Rather, these examples were chosen because the lecture is so commonly used and it contrasts so well with the Moore method. Further more, this dissertation is based on an adaptation of the Moore method.

Before proceeding with this discussion, it might be well to consider more precisely some of the terms being used.

Defini tions

Broudy defines teaching method to be:

. . . the formal structure of the sequence of acts commonly denoted by instruction. The term covers both the strategy and

'"Moore method" refers to the teaching style of the mathema­ tician Robert Lee. Moore. (The teaching style is described in later sections of this chapter.) R. L. Moore was born in Dallas, Texas, on November 14, 1882. He received B.S. and A.M. degrees from the University of Texas in 1901, and his Ph.D. from the University of Chicago in 1905. His doctoral research in the field of geometry was directed by Professor E. H. Moore. Further research by R. L. Moore in his speciality, plane analysis situs, contributed to the creation of a new area of mathematics now known as point set topology. After teaching at the University of Tennessee, Princeton University, Northwestern University, and the University of Pennsylvania, Professor Moore returned to the University of Texas in 1920 and has remained there since. He served as distinguished professor from 1937 to 1953, and his teaching career continued through the spring of 1969* In addition to teaching and his own research, Professor Moore has directed the Ph.D. research of a vast number of students. Many of these students made very significant contributions to the area of point set topology in their dissertations and have continued to have very productive careers as research mathematicians. Many of his students have also adapted his teaching style to their own courses and thus popularized the "Moore method." 3

tactics of teaching and involves the choice of what is to be taught as a given time, the means by which it is to be taught, and the order in which it is to be taught. [6: 3],

This quite extensive definition is the sense in which teaching method w ill be used in this study.

Waller and Travers describe the lecture method in this manner:

. . . one may note that many variations of the lecture are pos­ sible. Many teachers who consider themselves lecturers encourage questions on the part of students or ask questions themselves. About the most definitive statement one can make about the lecture method is that most of the time the instructor is commonly "talking to" the students. [41: 481],

A careful analysis of the lecture method is also given in Beard [4].

Specifically, in a mathematics class a lecturer presents the axioms, definitions, examples, theorems and proofs in a complete or nearly complete fashion. That is, most of the mathematics in the classroom is done by the instructor. The doing of mathematics by the students usually consists of reviewing the proofs and material presented and working on problem sets.

Moise describes the teaching method of R. L. Moore this way:

In three and a half years of graduate study, under his direction, I believe that I heard him lecture (in a style that a conventional professor would recognize as lecturing) for about two hours at most, and it may well have been less. All of his graduate courses worked in the same way: he would present postulates, definitions, and propositions; and the student's job was to find out which of the propositions were true, and present proofs or counter-examples. . . . At f ir s t, a very large number of the students' proofs were wrong. When a student had gone astray, Moore did not correct him; he would merely say that he didn't understand; and it took us quite a while to realize that anything that he "didn't understand" was surely wrong. Under Moore's regime, all use of the literature is forbidden. [23: 408].

Many of Moore's former students have adapted his teaching techniques to their own classes and in turn have influenced the teaching styles of their students and others who have directly or indirectly come in

contact with what has come to be called the Moore method. That is,

the Moore method is a method for teaching mathematics in which the

instructor supplies axioms, definitions and conjectures to the class

as needed. The students are then required to present proofs or counter

examples to the conjectures and hence develop the subject matter in a

deductive manner. The doing of mathematics in this case involves the

student in actually furnishing the proofs or counterexamples for all

the propositions on which the course is based.

It would be well to point out that although the Moore method

is based on the development of mathematics in a rigorous deductive manner from basic assumptions and definitions, it is not unique in

this respect. Indeed the lecture method is frequently used for the

same type of development. But the Moore method is an extreme example

of a mathematical method (not tied to a particular teaching method) which has become dominant to a large extent in this century— namely,

the axiomatic method. Allen describes the axiomatic method as:

. . . a method of exposition which furnishes a foundation for proofs as well as the principles needed for constructing proofs and for testing the validity of arguments. The foundation is provided by stressing the importance of carefully stated assump­ tions, properties, and definitions. The principles consist of such rudiments of logic as the transitive property of implica­ tion. [1: 1].

More on Moore

This dissertation is concerned with the exploration and

exposition of the writer's adaptation of the Moore method to an undergraduate class in real analysis and thus is not primarily concerned with the Moore method precisely as practiced by R. L.

Moore. However, a more thorough examination is valuable in that it uncovers several questions which are examined in this study.

The most valuable references for obtaining information about the Moore method are Wilder [*»3] and Moise [29], both former students of Moore. The following characteristics of Moore, his course and the Moore method are largely adapted from the afore­ mentioned sources.

1. The course in which the method developed was called

"Foundations of Mathematics" and dealt essentially with point set topology. Presumably the course content was that of the American

Mathematical Society Colloqium Publication by Moore [31]* Thus, as Moise states, the material was ". . . logically primitive, and fa irly isolated from that of other courses." [29: 409]-

2. Only students who were considered able to profit from the course were selected and the total number was kept small.

3. The course was graduate level, although an occasional undergraduate who had distinguished himself was allowed in the cou rse.

k. Rigorous proof was required with most of the intuitive material coming from the students themselves.

5. The propositions were not simple exercises. In fact, they sometimes turned out to be research problems which resulted in theses. 6. Students were required to work completely on their own-- no use of literature or outside resources was allowed.

7. Good natured competition between the students was encour­ aged. Under these circumstances, Wilder states: ". . . i t can hap­ pen that as many different proofs of a theorem w ill be given as there are students in the class." [^3: 479].

8. Dialogue between Moore and the class was an important part of the course. Wilder states that: "... he would encour­ age a student who seemed to have the germ of an idea, or put to silence one who loudly proclaimed the possession of an idea which upon examination proved vacuous." [^3: 480j.

9. The method was more relevant than the amount of material covered. According to Moise: "Moore's work proves . . . that sheer knowledge does not play the crucial role in mathematical develop­ ment that most people suppose." [29: 4093 - He continues:

The amount of knowledge that a small class can acquire, strug­ gling at every stage to produce its own proofs, is quite small. The resulting ignorance ought to be a hopeless handicap, but in fact it isn't; and the only way that I can see to resolve this paradox is to conclude that mathematics is capable of being learned as an a c tiv ity , and that knowledge which is acquired in this way has a power which is out of all propor­ tion to its quantity. [29: 409]-

10. Moore's scheme of teaching was highly successful. This

is evident by noting the large number of students who studied under him and went on to become distinguished research mathematicians— this, in spite of the fact that he was at a relatively isolated university and sometimes recruited his students from other areas of academic interest.

11. "Teaching in this style is not merely a negative matter of not lecturing; rather, it is a striking example of the art that conceals art, and it makes great demands on both mathematical and psychological depth." [ 2 9 : 409].

12. Similar methods have been initiated by others. In fact, there is evidence of international use of such methods.

Wilder reports [43: 479] that A. Tarski of the University of Warsaw has used similar methods in some of his courses, and MacLane

reports [27: 174] the same for the Japanese mathematician Oka.

Furthermore, certain elements of the Moore method--particu1ar1y dialogue between the instructor and student as described in char­ acteristic eight--bear resemblance to teaching methods used at

least since the time of Socrates and commonly referred to as the

"Socratic method." For a discussion of the Socratic method the

reader is referred to Broudy [6].

Some Questions

These points raise certain questions relevant to the adap­

tation of the Moore method to an undergraduate course in real analysis.

1. Is it necessary that the material be relatively unfamiliar

to the students? In particular, what happens when the in itia l mate­

rial is something as familiar as the real numbers? Wilder [43] reports having given a course in the structure of the real number system as well as a calculus course using the Moore method. He further reports knowledge of a course at the University of Miami which used a modifi­ cation of the Moore method to establish arithmetic on an axiomatic basis while leading into college algebra.

2. How does an average class (that is, a class in which the students are not selected prior to the course) respond to this method?

Wilder states that in large classes

. . . inevitably a few (sometimes only two or three) students "star in the production." I have found, however, that these "star" students often profited from having such a large audience as was afforded by the "non-active" portion of the class. Often the "non-stars" came up with some good questions and sometimes— rarely to be sure—with a suggestion that led to startling con­ sequences. [^3:482].

3. Is the Moore method adaptable to exclusively undergraduate classes? In particular, can undergraduates be sufficiently motivated to study mathematics in this manner? And if so, how much intuitive material must be provided by the instructor?

4. How w ill a student's concept of mathematics and his moti­ vation for studying it be affected by a course taught in this manner?

5. Can the Moore method be successfully adapted by an average

teacher of college mathematics? Or, is it the case, as Moise suggests,

that ". . . the method may require that the teacher be a genius"?

[29: 409].

6. What effect does this teaching method have on the creativity of undergraduate students? Both Moise and Wilder emphasize the role of

the axiomatic method in developing creativity. Yet there are those who

think otherwise. Perhaps one of the more vociferous of these is Morris K1 ine, who states:

In short, the axiomatic, deductive approach to mathematics omits the creative process, is often contrived, and is obscure. Exclusive use of it or even emphasis on it can be disastrous, especially for the training of mathematicians. [23: 58].

These questions w ill not be answered by this dissertation nor is it likely that they have definitive answers. However, they w ill be discussed with reference to the particular class involved in this study. In particular, this study is concerned with the exploration and exposition of this writer's adaptation of the Moore method to an undergraduate class in real analysis at Western Maryland College,

Westminster, Maryland, with particular reference to the members of the class. In order to take account of the variation in the students' backgrounds both as to the amount of mathematics they had taken and the teaching methods they had experienced, and the variation in their performance during the investigation, case studies are presented.

Origin and Background of the Problem

Throughout the writer's teaching career, he has tried to involve his students as much as possible in the class presentations.

This has led to considerable variation and experimentation with teaching methods, but usually involved a somewhat informal lecture

In which the students were encouraged to ask questions or make com­ ments freely and were in turn asked questions during the lecture.

Additionally, the students were usually expected to present solutions to homework problems during class. This way of teaching is probably typical of a great number of mathematics classes. It certainly reflects the writer's experience as a student. 10

During the summer of I9 6 7 , as a member of the National Science

Foundation Summer Institute for College Teachers at the University of

Georgia, the writer was enrolled in a course in real analysis taught by Professor G. G. Johnson. Professor Johnson used a teaching method similar to the Moore method, which gave the writer his firs t intro­ duction to such a method. This course proved to be very interesting and rewarding, and led the writer to consider trying the teaching method in one of his own classes.

Accordingly, in the fall semester of 1967, the writer used an adaptation of the Moore method in a linear algebra course. This hastily conceived attempt was not completely successful, but the successes and failures of this in itial tria l of a teaching method new to the writer motivated him to make the more careful and thorough study on which this dissertation is based.

Statement of the Problem

During the fall semester of 1969, the writer taught Mathematics

^03, Intermediate Real Analysis I , to a class consisting of eight junior and senior mathematics majors at Western Maryland College using i an adaptation of the Moore method. During this period, each student 1 kept a diary of time spent, problems worked, and his occasional reac­ tions to the course and teaching method. The writer also kept daily records of his and the students' reactions and progress. Additionally, each student was interviewed privately during the course and again after completion of the course. Based on this collection of information, the problem was to explore this teaching method in conjunction with case studies of the students participating in the experimental course. There are two phases of the problem which are investigated in this dissertation. First, the writer has attempted to provide a qualitative description of the particular method (including course materials--see Appendix A) used with these students this semester as

(i) a method for teaching upper division undergraduate mathematics, and (ii) a method for teaching real analysis. The emphasis here is on the exploration of a teaching method.

Secondly, the writer has examined the assumptions that this teaching method will affect (i) the student's concept of the nature of mathematics, and (ii) his approach toward learning mathematics.

Specifically, it was hypothesized that upon completion of the course

the student would:

1. Consider mathematics as more than numerical problem solving.

2. See more clearly the role of axioms, definitions, proof

and, most particularly, counterexample.

3. See that a variety of proofs and alternate approaches are

possible in developing mathematics.

4. Gain confidence in his own creative powers and see mathe­

matics as a creative process which he can do.

5. Depend more on himself to learn.

6. Not devote his study time to memorization for recall on

tests, but rather to being creative.

7* Experience a sense of accomplishment which would motivate

him to do more independent work.

The case studies were of primary importance in seeking evidence of

these effects. 12

Although less formally, certain other points were being kept in mind during this investigation. One of these concerned the class­ room role of an instructor using the Moore method. The classroom presence of such an instructor is certainly less conspicuous than that of a lecturer. But does this mean that he has lit t le to do with the progress of the class? What influences do his comments or lack of comments have on the students?

Another point concerns learning to teach. The writer has previously commented on factors shaping his teaching style. Is an individual's method of teaching something that evolves through trial and error? More generally, howdoes a teacher learn to teach?

These points are not the kind of things for which specific data were collected. They w ill be at most obliquely examined in the body of this study, but w ill reappear with the writer's impres­ sions in the final chapter.

Review of Related Literature

Several references to related literature have already been made and others will occur later. Literature specifically relating to a study of this type is not extensive, nor is literature relating specifically to the Moore method. Thus, this section is primarily concerned with representatives of literature which bear only a periph­ eral relationship to this study. An attempt was also made to restrict the literature cited, as much as possible, to that referring to col­ lege teaching. In particular, no mention is made of the many studies dealing with discovery teaching in school mathematics. R. L. Moore has never written about his teaching method, although he has described his method and ideas in a film produced by the Committee on Educational Media under the auspices of the

Mathematical Association of America.^ However, the two articles previously cited by Moise [29] and Wilder [43], both of whom are distinguished former students of Moore, provide excellent exposi­ tions and analyses of the Moore method.

Although this study is not involved with a comparison of the Moore method and some other method, it is interesting to look at a few studies of this type. A good example of a study along

these lines was carried out by Guetchow, Kelly and McKeachie [19] at the University of Michigan. They compared recitation-, discus­ sion and tutorial methods in teaching freshman psychology and found few significant differences among the teaching methods with

respect to the students' learning of psychology. This conclusion seems to be typical of most studies which compare lecture with discussion in college teaching. Waller and Travers [41] give a

long lis t of such studies and although a few studies report some

significant differences between the methods (some claiming lecture

superior; others discussion), the relevance of the variables seems

to be subject to question. Therefore, Waller and Travers conclude:

". . . most studies find no significant differences between lecture

^This film entitled "Challenge in the Classroom" is dis­ tributed by Modern Learning Aids Division of Modern Talking Picture Service, Inc. 1212 Avenue of the Americas, New York, New York 10036 . and discussion methods." [41: 481].

Related studies involving lower division college mathematics teaching have been done by Cummins [11], Ernst [13], and Filano [14].

Filano examined the effectiveness of a method in which a minimum of one-half of the class time was given to student participation as opposed to listening and watching in the control group. He found significant gains in the performance of the experimental group in college algebra; but no significant difference in the analytic geo­ metry groups. Cummins, in comparing a student-experience-discovery approach to the teaching of calculus with traditional instruction found that the students in the experimental classes did as well on problems and manipulative skills, but had increased understanding, a superior knowledge of the fundamental theory and logical relations of calculus, and had experienced the "thrill of discovery." Ernst's

study was primarily an exploratory study of a teaching method which

involved student discovery and class discussion in a class of high- ability university freshmen with a previous record of low achieve­ ment in mathematics. Ernst designed a multiple choice test to measure

the students ab ility to read, understand and apply relatively unfamil­

iar material. He stated that the experimental class performed much

better on this test at the end of the quarter than they did at the

beginning.

In a different vein, literature not specifically related

to the Moore method but seemingly in support of the same ideals

comes from both psychologists and mathematicians. For example, 15

Bruner states: "There is nothing more central to a discipline than its way of thinking; it is an epistimological mystery why traditional education has so often emphasized extensiveness and coverage over intensiveness and depth." [7: 1013].

Hatch, in an examination of independently prepared statements by Robert Gagne, Ralph Tyler, and W. J. McKeachie for practical impli­ cations for college teaching found these points concerning good con­ ditions for learning:

"The human learner . . . is made the central part of education as a system." (Gagne)

The learning reflects that which "the learner learns," that is that which "he is thinking, feeling, or doing." (Tyler)

The learning is "active" rather than "passive." (McKeachie)

"The learning situation encourages generalizabi1ity, the learn­ ing of principles, as opposed to . . . rote learning." (Gagne)

A "principle" is learned "in a new situation." This helps one to "identify the common element in situations and shortens the learning process." (McKeachie)

A student "explores something new." (Gagne)

"Each new practice requires him to give attention to it because of new elements in it . . . [only so] does it serve adequately as a basis for effective learning." (Tyler)

Importance is attached to "levels of aspiration." (Gagn£)

The learner "sets high standards of performance for himself . . . high but attainable." (Tyler)

"We can teach students to enjoy learning." (McKeachie) [21: 7“8].

The need for courses at the undergraduate level which "let students in on what mathematics is all about" is frequently stressed by mathematicians. For example, Andrew Gleason commenting on training 16 for graduate study states: "The primary goal of undergraduate train­ ing is not knowledge but savoi r fai re. One aspect of savoi r fai re is the ability to read and write m a th e m a tic s [18: 32k] . Later on, he says: . . our fundamental goal: To teach what mathematics is and how it works." [18: 925].

Some of the same ideas appear in a different context when

Kenneth May discusses undergraduate research, which he says refers

. . . to an educational activity designed to increase the amount of creative mathematicians and scientists by giving young people the experience of original work in addition to the standard courses where they "learn" what has been discovered by others

. . . this kind of activity shades off into . . . pedagogical techniques not usually called undergraduate research. Examples are the "Moore method," contests, and the "discovery approach" at all levels. The inspiring teacher who stimulates his students to work on their own and solve challenging problems is promoting undergraduate research. [28: 70 ].

Andre Weil puts it this way:

The student should therefore be gradually accustomed, by means of startling examples, to question the truth of every unproved proposition, until at last he is able to deduce from the ordinary axioms everything he has learned.

The teaching of mathematics must be a source of intellectual excitement. This can be achieved, at the later stages, by taking the student to the brink of the unknown; at earlier stages, by making his solve for himself questions of theoretical or practi­ cal importance. [^2: 35].

And fin ally, general support for a study of this type comes from a quote by Beard:

. . . carefully planned sequences of experiments or inquiries are s till needed. It is not usually possible to draw general conclu­ sions from comparison of one teaching method with another on one occasion in a single department. What is needed is a concerted effort in studying each method, collating information already available, and experimenting with variations of the method to see which ones are most effective and under what circumstances. [4: 5^]. Design of the Study

The course

Mathematics *f03, Intermediate Real Analysis I, is the firs t course in a possible two-course sequence in analysis taught by the writer at Western Maryland College during each academic year. As a one-semester course, Mathematics *»03 is usually offered again in the spring semester by a different faculty member. Although the only prerequisite for Mathematics 403 is completion of the three-course elementary calculus sequence, students usually take the course their senior year. For the seniors involved in the study, the course was

recommended, but not required, for graduation. About two-thirds of this senior class elected the course during one or the other semesters of 1969-70.

The students

The class involved in this study consisted of eight mathematics majors at Western Maryland College. The composition of the class is

illustrated in Table 1 on the following page. The column Sem.Hrs.Math,

refers to the semester hours of mathematics the students had completed prior to the fall semester of 1969-70 . Mathematics courses through calculus III were considered to be introductory courses. They are

reflected by the difference in the subcolumns Total and Beyond Cal.

The hours beyond calculus perhaps better indicate the students' backgrounds. 18

TABLE I

CLASS PROFILE

Student Year Sex Math GPA Sem.Hrs.Math. First Number 3.0 Base Total Beyond Cal. Course

1 Senior Female 2.00 21 15 Cal. II

2 Senior Male 1.92 25 13 Anal.Geom.

3 Junior Male 3.00 15 3 Anal.Geom.

k Senior Male 2.32 31 19 Anal.Geom.

5 Senior Male 2.89 28 19 Cal. 1

6 Senior Ma le 0.76 33 18 Alg.-Trig.

7 Senior Male 3.00 22 13 Cal. 1

8 Sen ior Male 0.39 28 13 Alg.-Trig.

As is probably evident from the table, the seniors were not

selected; rather, they were allowed to enroll in the course of their

own volition. Four of the better junior students were invited to

enroll in the course. However, only the one junior chose to enroll.

Examination of the students' backgrounds revealed that several

of them were weak at writing proofs. (Individual cases are discussed

in Chapter I I I . ) Thus, it was not expected that all of the class

would be won over by this teaching method. Indeed considering the

late stage of their undergraduate mathematical development, it was

considered likely that some of the students would make lit t le , if

any, progress. 19

Data Collected

Daily records were kept by both the students and the instructor

(writer). The students were given a duplicated booklet at the beginning of the course with instructions to keep a daily record of time spent on the course and the problems they were working on at that date. Space was also provided for optional candid remarks the student might have on that date. In addition to the daily record keeping, the students were instructed to write substantial expressions of their reactions and feelings at three or four times during the course.

The instructor kept records which included individual daily records of the performance of each student and a daily record of the class as a whole including the instructor's feelings and reactions.

Furthermore, each class session was tape recorded and the tape was reviewed for relevant comments.

In addition to the record keeping, each student was interviewed privately by the instructor--once during the course and again after completion of the course. The midterm interviews were conducted the week of October 20-2k, 1369; the final interviews were from January

5-9, 1970.

Prior to the beginning of the course, the writer obtained information about each student's background which included the courses he had completed with as much information about the courses as possible and comments from his instuctors on his ab ility to write proofs.

The case studies in Chapter III are based on the information 20 collected from these various sources. Guidelines for interviews, forms for data collection and directions to the students are collected in Appendix B.

The Teaching Method

The students were given duplicated notes a few pages at a time throughout the semester. The notes contained axioms (when needed), definitions and problems. Here "problems" is used in a broad sense-- a problem might be a question, a proposition or conjecture, or it might direct the student to perform some activity such as make up some examples or formulate a definition. The course materials and diicot’ ,s to the student concerning them are given in Appendix A.

The students were instructed to try to obtain solutions to the problems outside of class. The college operates under an academic "honor code" and the students were put on their honor not to consult outside resources for a solution.

It is worthwhile to point out that usually the writer encour­ ages his students to work together on their homework assignments. It is fe lt that the sharing of ideas which occurs under these circumstances is beneficial to both the students having d ifficu lty and the more able students. However, for purposes of this study, the students were put entirely on their own. That is, it was considered a violation of the honor code to give or receive help on a problem prior to its presenta­ tion in class. Sharing of ideas after the presentation were encouraged.

Class activity usually consisted of students' presentations of their solutions. Occasionally a student was allowed to present the 21 solution for a problem as a volunteer. But a student was usually called on by the instructor in such a manner as to provide a sharing of the opportune ties. A student's solution was open to the scrutiny and criticism of his classmates and the instructor. In case of chal­ lenges, the student was allowed to make corrections or additional explanations. If a solution was irreparably wrong, the student was given an opportunity to present a new solution at the next class meeting. Students having alternative methods of solution (when applicable) were allowed to present them voluntarily. Each student was graded on his contributions and his final grade came primarily from his classroom contributions, although an hour test was given near the end of the course.

Course Content

The topics chosen for the course content reflect the writer's courses in analysis as a student and as a teacher. No claim for orig­ inality is made, other than possibly the order of presentation. An attempt was made throughout the course to have propositions of widely varying difficulty. That is, the poorest student should occasionally find a problem he could work and the best student should occasionally have difficulties.

Chapter 1 of the course notes begins with very elementary material which was intended to build the students' confidence in their ability to write proofs. Although this introductory material may seem to some people to be triv ia l, it was considered to be worthwhile in view of the students' backgrounds. Beginning with section five, the 22 problems become more sophisticated. Section six includes versions of both the Heine-Borel and Bolzano-Weierstrass theorems. Sections seven and eight review the ideas of continuous function and derivative from elementary calculus, but introduce them geometrically. The writer was introduced to this approach by Professor Johnson of the University of Georgia. Section nine introduces function sequences and includes a problem about equicontinuous fam ilies of functions.

Chapter 2 of the course notes introduces a d iffe re n t level of abstraction and the language associated with i t . I t was intended

that many of the topics of chapter 1 would be redone in chapter 2 with greater generality; thus certain features of the "spiral approach" would be incorporated into the course. However, there was not sufficient time to do more than the introductory section of chapter 2.

Organization of the Dissertation

This first chapter has provided an introduction and overview of the study including a description of the Moore method and questions

raised concerning its applicability to teaching undergraduate real

/ analysis. Chapter II w ill deal with the course as a whole. Examples

of classroom a c tiv ity and the w rite r's feelings about the course w ill

be included. Chapter I I I w ill be concerned with the individual stud­

ents involved in this study. Chapter IV w ill examine the questions

concerning the Moore method raised in Chapter I in lig h t of the material

in Chapters II and I I I . Informal conclusions, more reactions of the writer, and further questions will be given in Chapter V. CHAPTER I I

THE COURSE

This chapter gives an overall view of the course. Progress through the course m aterial is chronicled, examples of student a c ti­ vity are presented, and the instructor's reactions are expressed.

These comments are based prim arily on records and notes made by the writer during the course.

Progress Through the Course

Introductory material

As may be seen from the course notes (Appendix A) sections one through four dealt with familiar properties of the real numbers.

Section one was concerned with properties of addition; section two with properties of multiplication; section three with properties of order; and section four with properties of absolute value. However, even though fa m ilia r, this was the f ir s t time many of the suudents had been required to prove these properties based on only a short lis t of axioms. At the beginning there was frequent skepticism as • to whether things such as -0 = 0 could really be proved. This skepticism was compounded when some of the "proofs" presented made unnoticed use of other unproved properties. For example, in the first proof presented for a • 0 = 0 , the student wrote the additive inverse of (-a)( 0) as -(-a)(0) , then la te r wrote this as a • 0 by 2b using, he claimed, the property proved earlier that -(-a) = a. It took considerable prodding before another student discovered the error.

The skeptics were eventually made into believers when correct proofs were presented for the problems.

After the f ir s t class period which involved handing out dupli­ cated material and discussing the format for the course as well as the only real lecturing by the instructor (set operations and index sets were presented with complete, precise proofs by the instructor), parts of seven class periods were devoted to the introductory m aterial.

Each member of the class participated to a considerable extent during this time. There was exposure to a variety of proofs including a number of proofs by the indirect method. The students seemed to have had very little previous experience or opportunities to use the indi­ rect method. There frequently was collaboration on proofs in class-- that is, when an error was discovered in a proof, another student would be able to correct it so that together a correct, rigorous proof would be presented. Alternate proofs were also frequently presented—these often involved interchanging the order of the prob­ lems as given in the notes. Several of the problems for which every­ one seemed satisfied with their proofs were omitted.

Section five

The problems in section fiv e , which concerned properties of positive integers, began to demand an increased sophistication from the students. As a consequence, the number of students able to achieve and participate actively in the class began to decline. For the f ir s t time there were problems that no class member was in i­ tially able to present. This necessitated giving hints for the first time in the course. For example, problem 1.17 (Math Induction) was brought up the seventh class period. When no one was able to present i t, the following hint was given: " I f S is a subset of and S is a successor set, then S = N_. Let S = {n | Sn is true}." This prob­ lem was discussed further the next class period and a joint proof was arrived at as a result of the discussion. A member of the class was assigned to write up the proof precisely and have it duplicated for the class members.

The limitations of mathematical induction were discovered when a student tried to use induction on the number of elements in a nonempty set of positive integers to prove the wel 1-ordering p rin cip le, problem 1.18, during class number eig ht. Several hints for problem 1.18 were given, including two in the form of lemmas, before it was eventually talked through four class periods after the initial discussion.

Section five, then, was somewhat of a turning point in the course. The class began to separate into the students who could achieve although sometimes frustrated and those who were to face a great deal of frustration, with little in the way of visible achievement for the remainder of the course. The number of hints given during this section also produced an effect. It seemed that some of the students began to wait for a hint on each problem.

Therefore, after the twelfth class period fewer and more vague 26 hints were given.

The following remarks express the writer's feelings at this point. They were w ritten on September 29, a fte r the eighth class period.

Certainly the classes to this point have been interesting if not illum inating. Good work has been done, but there seem to be signs of growing fru s tra tio n . We are ju s t now getting to some significant material and I am somewhat apprehensive about, the mood o f the class. There seems to be some tendency to just "shut off" on a difficult proof. Also, I may have to cut down on the number of h in t s - - it seemed today that they were just waiting for me to give new hints.

Section six

Parts of class periods nine through sixteen were devoted to section six which was concerned with properties of completeness.

This was probably the most important period of time for the class.

It was the beginning of the study of mathematical analysis and as a result the first unfamiliar material for the students. This was a very exciting period although there were low points in class morale and high points in fru stratio n during this time. However, by the time the section was completed, there seemed to be a new positive attitu d e emerging from certain members of the class. They had been challenged, had met the challenge successfully, and now were confi­ dent of their ability to meet future challenges. Some excellent work of considerable depth had been attained by several students.

To borrow a phrase from W ilder, these students were to "s ta r in the production" for the remainder of the course. There was also some evidence of Increasing sophistication among class members making fewer presentations. They began to show an increased aware­

ness of what constituted a proof. That is, they were sometimes able

to make suggestions for proceeding in proofs which stalled. Also,

they began to question the rigor of some of the presentations.

Another ch aracteristic which continued throughout the course

first became evident during this period. That is, a student would

present an incomplete proof or maybe just state that he had tried

a certain approach. Given encouragement that there was the germ of

an idea there, the student would proceed to develop i t and present

a complete proof at a later class period.

Problem 1.25, the existence of the square root of two, was

o rig in a lly discussed on the eleventh class meeting and went unproved

u n til the tw enty-third class meeting, a period of four weeks. This

was the longest any problem that was eventually solved remained

unsolved. This time lag apparently upset many of the students as

several complained about i t during midterm interviews.

Perhaps the low point in class morale came with problem 1.27

the Heine-Borel property. This was probably the f ir s t problem which

was not intuitively obvious. It was brought up during the twelfth

class meeting. When no proof was forthcoming, examples of coverings

of sets were given and this hint: Let S ■ {x | [a,x] has a finite

subcover}. The theorem and this hint seemed completely foreign to

the students. It is doubtful that it would have been proved if one

of the students had not recalled the proof from topology. This

student was able to reconstruct the proof, and after checking with the instructor presented it to the class, taking a full class period to do so. This was the first significant result at this level of abstraction and the class reaction was quite negative. The attitude seemed to be "I can't do proofs like that, so I have no intention of trying problems lik e that, and you should not expect us to try such problems." It is doubtful whether any of the students understood the proof or believed it at this time. However, after the initial shock wore o ff, it seems that the students came to understand the

Heine-Borel property and made use of it to present elegant proofs in la te r problems. An alternate proof of the Heine-Borel theorem using the Bolzano-Weierstrass theorem, problem 1.31» was given by the instructor at a la te r date. This approach to the proof seemed more acceptable to the students.

Picture "proofs"

After the quite abstract arguments of section six, the intuitive, geometric approach to continuity and derivatives in sections seven and eight seemed to appeal to the students. Class periods seventeen through thirty-three were spent of these sections which were considered to be the heart of the course. The tempo of the classes was quite variable through these sections. The students would appear to struggle m ightily with the problems, break through with a rush of resu lts, then become stumped again.

A frequent source of trouble in the presentations was in relying too much on pictures for the proof. This resulted in a number of erroneous presentations. However, this was quite valuable, for just as a picture might convince a student that he had a proof, another picture could demolish his argument. An example of this occurred during class period th ir ty . Problem 1.56 states:

I f f is a simple graph with slope m > 0 at (x,y) , then there exist two vertical lines \lx and V2 with (x,y) between them such that if (s,t) t (x,y) is a point of f between Vj and V2, then f(s) < f(x) if s < x and f(x) < f(s) if x < s.

To prove this, a student drew a picture (Figure 1) and arguing from the picture gave what appeared to be a very straight-forw ard proof based on the intersection of the graph with the tangent lin e. (Take

(x,y)

V V 2

Fig. 1.—Picture proof

Vj and V2 at the points of intersection, if they intersect; if not, take any two vertical lines.) No one challenged this "proof" so we moved on to other problems, but left the picture on the board.

Near the end of the period, another student asked if we could return to problem 1.56 as he thought he had a picture which would provide an exception to the "proof." His picture failed to be an exception, but now the student who o rig in a lly had presented the problem found an exception (Figure 2). The following class period, the student

/

S' (x.y)

Fig. 2.—An exception making the erroneous presentation presented a rigorous, deductive proof of 1.56 without drawing any pictures.

This behavior occurred several times. It seemed that after

"getting burned" a student would become quite p ro ficien t at separat­ ing in tu itio n from fact. The members of the class were usually quite willing to criticize presentations based on drawings.

These two sections were probably the best for developing

"mathematical maturity." In addition to experience with pictures and counterexamples, the students began to learn to simplify their work by using "finesse functions." (That is, ways to extend results by defining appropriate functions. For example, problem 1.49 can

be done elegantly by defining g(x) = f(x) - y and applying problem

1.48.) Thus, in addition to just being able to work a problem,

some of the students began to look for ways to shorten or simplify

the proof.

As previously stated, the amount of material covered during

a given class period was quite variable. For example, during class

period twenty-eight only problem 1.53> the uniqueness of slope, was

discussed. In this case, four different proofs were presented and

considerable class discussion developed concerning the merits of each.

Two problems in these sections were not solved. Problem 1.52 was made an optional hand-in assignment and a reference [5] was given

for i t . The chain ru le, problem 1.59v, was also unsolved in spite of

considerable prodding by the instructor.

The remaining material

The students seemed to tire the last few weeks and their per­

formance was more e rra tic . Three class periods were spent on sequences

with only two students presenting problems during this time. The

The Cauchy Criterion, problem 1.64, was not solved completely, nor

were problems 1.68 and 1.69. A "proof" of problem 1.66 was given

during class period thirty-six. One student questioned the role of

the 6's in the proof but withdrew his objection the next class period.

Without explaining why, the instructor gave the class two examples of

function sequences to consider. It wasn't until five class periods

later that anyone discovered these to be counterexamples to problem 32

1.66. This discovery was made by the student who presented the original proof.

The last two weeks of the course were spent on the metric space properties in chapter 2, section 1. Perhaps due to the more elementary nature of the material, a larger number of students par­ ticipated in this development. A very elegant indirect proof of the

Swartz inequality, problem 2 .4 , was presented.

The hour test

The hour test was given during class period thirty-nine on

December 12. A copy of the test is included in Appendix A. The test was intended merely to provide a check on the students and a certain basis fo r comparison with the previous year's class. Percentage scores on the test were: 93, 86, 81, 77, 75, 60, 40, 39* With two exceptions, these scores were in the same rank as the students were placed on the basis of their overall performance. Only one proof was required on the test. Although the definition of limit had not been given previously in the course, the required proof was similar to those given in class fo r the analogous properties of continuity and derivatives. The proofs given by five of the students were judged satisfactory. The proofs of the remaining three students were con­ sidered to be unsatisfactory.

Daily progress

The table on the following two pages charts the day to day progress of the class. Frequently problems were discussed in class 33

TABLE 2

DAILY PROGRESS

P rob 1ems Students ri< Di scussed Presenting

1

2 . 1, 1.2 1, 2, 3, 4, 5, 6, 00 oo r^. 3 •3, 1.5, 1.6 1, 2, 3, 4, 5, 6,

4 .6, 1.7 1, 5, 8

5 . 8, 1.9 , 1.10 7, 8

6 .8, 1.9, 1.12, 1.13 1, 2, 3, 4, 6, 7,8

7 .13, 1.14, 1.15, 1.16, 1.174, 5, 6, 8

8 .13, 1.17, 1.18, 1.19 1, 2, 5, 7

9 . 18, 1. 19 , 1. 20, 1.21 4, 8

10 . 17, 1. 18, 1.2 1, 1.2 2, 1.23 5

11 .18, 1.24, 1.25, 1.26 5

12 .18, 1.23, 1.25, 1.27 7

13 .27 5

14 .27, 1.28, 1.30 3

15 .27, 1.28, 1.29, 1.30 3

16 .31, 1-32 5, 7

17 .33, 1-34, 1.35, 1.36, 1.373, 4, 7, 8

18 .32, 1.38 2, 3, 5

19 .38, 1.39, 1.40 2, 5, 7

20 .40 3 , 8

21 .41, 1.42 34

TABLE 2— Continued

Problems Students ri< Di scussed Presenting

22 1.41, 1.42, 1.43 5, 7

23 1.25, 1.43 2 , 4

2 4 1.44 5

25 1.44, 1.45, 1.46, 1.47, 1.48 2 , 3, 5, 7

26 1.48, 1.49, 1.50, 1.51, 1.52 3, 7

27 1.44, 1.49, 1.51, 1.52 3, 7

28 1.53 3, 5, 7, 8

29 1.54, 1.55, 1.56 1 , 4, 7

30 1.55, 1.56, 1.57, 1.58 3, 4, 5

31 1.56, 1.57, 1.59 1 , 3, 5, 7

32 1.59, 1.60 5, 7, 8

33 1.61 , 1.62 5

34 1.63, 1.64 5, 8

35 1.59, 1.65, 1.66 2 , 7

36 1.67, 2.1 1 , 5

37 1.68, 2.2, 2.3, 2.4, 2.5 1

38 1.66, 2.3, 2.4, 2.5 3, 5

39 (Hour test)

40 2.7, 2.8, 2.9 1, 3, 7

41 1.64, 1.66, 2.9, 2.10 5

2 .10, 2.11 5 over several periods before a solution was arrived at. Also, some­ times alternate solutions or modifications in the hypothesis of a problem were discussed in succeeding class periods. These a c tiv itie s result in the Problems Discussed column of the table frequently showing the same problem for several days. Only students who actu­ a lly made eith er the in itia l or an alternate presentation (perhaps erroneous) of a problem are included in the Students Presenting column. This does not re fle c t the many valuable comments that might have been made by other students during the class session.

Conduct of the Classes

The classes were held in a small seminar room where a ll class members, including the instructor, were seated at tables.

An atmosphere of inform ality and freedom of expression was encour­ aged .

The classes were real learning experiences for a ll concerned.

Often the work of one student would result in another student get­ ting ideas about a problem with which he had been having d iffic u lty .

Thus proofs were sometimes partially constructed during the class sessions. This also led to occasional collaboration on proofs during a class session. That is, one student would start a proof and realize that he had made an error at some point. However, another student who may have not been able to begin the proof would see how to pick it up at this point and carry the proof fu rth er. At times several students would p articip ate in this production with perhaps the original student putting the finishing touches on i t .

During the class sessions there was usually a certain amount of byplay between the instructor and the class members.

Sometimes it was necessary to conjole, wheedle, or even harrass certain students to bring out comments or presentations. The class as a whole was the ultimate judge as to whether a proof was accepted or rejected. The members of the class were usually quick to point out obvious errors in proofs. However, less obvious errors were sometimes overlooked by the students (and probably the instructor as well). In these instances, the instructor's expressions and questions usually conveyed the idea that some­ thing might be wrong. On a few occasions, proofs that were par­ tia lly incorrect were let stand when the class accepted them.

The students were informed that this was a possibility.

The instructor presented a few proofs during the course.

At times when a proof had been talked through by the class, the instructor would then write up what he considered to be a precise, rigorous proof of the statement. Alternate forms of a proof were also presented on occasions when i t was f e lt that they would contribute to the understanding of the material. In either case, the instructor's proofs were open to the same c riticism as those of the students. Usually they received it. Reactions of the Instructor

The classes

One of the most interesting parts of the course fo r the w rite r was the opportunity to see new and unexpected approaches to the proofs of theorems. Thus, the class sessions were always something to look forward to. This enthusiasm fo r the unexpected seemed to be shared by the students as well. There seemed to be an air of anticipation not found in other classes in the w rite r's experience. When a class had gone extremely well the stim ulation and excitement from i t tended to a ffe c t the w rite r long a fte r the class period had ended. Results from i t were frequently discussed with colleagues and unsuspecting students who happened to come by. Discussion of a result with one or two of the class members frequently continued beyond the class period also.

There were classes that did not elicit such positive reactions, however. At times the class really struggled with the problems.

During these periods when there seemed to be little achievement, the course seemed to drag, and a certain amount of frustration on the part of both the students and the instructor was evident. The frus­

trations of the students are evident at times in the case studies of

the following chapter. Some of the frustrations of the instructor will be evident in the following paragraphs.

As is probably evident, the writer was quite emotionally

involved with these students— perhaps more so than in any previous

teaching experience. This involvement resulted in wide ranging emotions and feelings about the course that frequently changed from 38 one class period to the next. The variety of these feelings is expressed in the following quotes taken fo r consecutive class meet­ ings from the class record kept by the w rite r. The dates were from

November 6 through November 19.

Class 24: The students seem reluctant to do the 6 - e adjust­ ments necessary fo r the proofs to work out elegantly. They also seem to f a il to understand the importance of 6 .

Class 25: This was one of the best classes in several days.

Class 26: There was considerable class discussion today—more so than usual with comments even from students who do not usually participate. It has been hard to find time to reflect on the course lately, but actually it has gone well the last two periods. A fter the hang-ups with e and 6 , class has proceeded more smoothly. In retrospect, we have covered a sig n ifican t amount of m aterial, completing the usual properties of continuity today. They now have the rest of chapter 1. It seems possible that we will com­ plete it and possibly some of the other material before the end of the semester. Although the course has been slow and draggy at times, I would rate it as a success at this point.

Class 27: This was an interesting class with the students involved in a d iffe re n t manner than usual. Lots of work was constructed on the spot with a resulting conflict of ideas.

Class 28: In spite of the fa c t that only one problem was covered, the class, after the first few minutes, did not seem to drag. In fact, it was quite interesting.

Class 29: My mood must be darkening again. This was a very unen- thusiastic class (myself included).

Class twenty-nine was so bad, in fact, that i t precipated these more disparaging remarks written the evening of November 17-

After an upturn in attitude last week—there were even some stim­ ulating discussions—today was blue Monday again. The class seems very draggy and the students apathetic. The work that was pre­ sented today was very discouraging. The proofs were poor— or I should say the attempts at proof since one of the problems was actually false. It seems that they haven't re a lly learned any­ thing about w riting a proof— they ju s t blunder in with no idea of what they have or where they are going. The failure to even 39

see what must be shown and the tendency to just draw a figure and show a picture for that particular curve is particularly appalling. For example, a student "proved"(a 1 though he admitted that it was not rigorous) that a differentiable function is continuous. He drew a curve on the board and proceeded to draw several horizontal, s la n t, and v e rtic a l lines on i t meanwhile expounding about how if these vertical lines didn't work you could just pick some in closer. No mention was made about what he needed to show and no method for finding the vertical lines other than drawing them was given.

But what a difference a day makes.

Class 30: This was a rather great class. Very l i t t l e in the way of material was covered, but it was one of the better classes for excitement. There were several good comments a ll around the room. There was more involvement from more students than is frequently the case.

To the credit of the students, they continued to work through­ out the course whether the mood was high or low. In fact days when l i t t l e was accomplished perhaps stimulated some of the students.

Rather than experience another fru stratin g session they took i t upon themselves to get some work done fo r the next class period.

The interviews

The midterm and fin a l interviews averaged about one hour in length each time per student. They were conducted in the w rite r's o ffic e with the door closed to insure the privacy of th e ir remarks.

The students were asked to react candidly to the questions. The extent of th e ir comments was fle x ib le and they could decline to comment on any question, i f they desired. Furthermore, they were encouraged to digress, go off on tangents, and interject other comments or questions that arose during the interview. In fact, the interviews were con­ ducted with such an air of informality that they might more appro­ priately be called discussions. Other students, other courses, and other teachers were frequently mentioned. The degree of closeness ko and good feeling that existed between the instructor and students le f t no reason for d is tru st. Thus, the w rite r feels that the com­ ments made by the students were candid and honest.

The midterm interviews occurred at a time when the class was struggling. Perhaps as a result of this, the keynote of these inter­ views seemed to be frustration. The students generally reacted favor ably to the course and teaching method, but frequently prefaced th eir remarks with comments about how fru stratin g it was. There was also a feeling that the course had not re a lly progressed fa r enough to form d e fin ite opinions.

For sp ecific purposes of this study, there was l i t t l e of value in the midterm interviews. However, they were f e lt to have been worthwhile from a pedagogical viewpoint. They provided an opportunity for the instructor to sit down with each student indi­ vidually, get better acquainted with him, and talk about the course.

Under these conditions, the instructor was able to encourage, reassure, or chide each student in a manner not possible in the classroom. On the other hand, each student had a chance to express his feelings, gripes, frustrations, or whatever. In short, to use an overworked word, there was an opportunity fo r the instructor and each student to communicate.

The mood at the final interviews was quite different. Maybe i t was ju st due to r e lie f that the course was over, but there were very few expressions of feelings of frustration with the course. It seems unlikely that the students were concealing their frustrations or negative feelings. Theirgrades had been assigned prior to the interviews so whatever they said would have no effects gradewise.

It seems likely that there was an adjustment to the frustrations as the course developed. That is , i t took a certain amount of time

for the students to adjust to this teaching method and the emotions

associated with it.

There was evidence of boredom with the final interview on

the part of some of the students. They seemed to think that they

had said it all in their diaries or the midterm interview.

Overall reaction to the course and teaching method was mixed.

The reaction of four of the students might be termed positive. That

is, they were enthusiastic about the course, thought it had been a

success, and thought the teaching method should be continued. Three

of the students seemed neutral toward the course and teaching method.

One student expressed disappointment with the course.

Summary

This chapter has dealt with the progress of the students

through the course m aterials and th e ir classroom behavior. I t was

seen that as the semester progressed, fewer students chose to par­

ticipate in the presentation of solutions to the problems. S till

there seemed to be an increased awareness on the part of all the

students as to what constituted a proof. Even some of the students

who seldom presented results became quite successful at criticizing

the proofs being considered. The learning experiences encountered

in the classroom were of value in themselves. It seems, then, that

the Moore method might well be called a learning method. The close involvement of the instructor with the class resulted in a variety of feelings from day to day about the course and students. Overall, though, the instructor was pleased with the course and the performance of the students.

Case studies of the students will be given in the follow­ ing chapter. Each student's background p rior to the real analysis course is examined as well as his performance and reactions during the course. CHAPTER 11 I

STUDENT PERFORMANCE

This chapter presents studies of the individual students involved in the real analysis course and this investigation. Each study is in two parts. The first part introduces the student and deals with his academic background while the second part describes his performance in the course.

The data for the background portions was collected during the spring and summer of 1969- The records of the Mathematics

Department at Western Maryland College were consulted to supplement the w rite r's personal knowledge of the students. For the most part the background portions were w ritten p rio r to the course. Thus certain pre-judgements are evident in them.

As part of the investigation of the students' backgrounds, a complete 1isting of mathematics courses and grades for each student was compiled. In order to go beyond the mere listing of courses, the writer delved further into the method and content of each mathe­ matics course for the particular term it was taken. When possible, the course instructors were contacted personally. From these discus sions and departmental records, descriptions were prepared which stated the name and number of the course, the instructor, the year and semester offered, the textbook used, and the students from this

*»3 investigation who took the course. When available, each description also included the instructor's teaching method, course outline and final examination. Frequently, the writer also examined other tests, problem sets, and duplicated materials prepared by the course instru- to r. Samples of these descriptions are given in Appendix D. Catalog descriptions of a ll mathematics courses involved in this study are given in Appendix C. The instructors contacted were also asked to com ment on the proof writing ability of the students involved in this investigation on the basis of their performance in the instructor's courses. Mathematics course listin g s for each student together with his instructors' comments are included in Appendix E.

Certain common elements in the students' backgrounds can be identified. Based on the course descriptions and contacts with the instructors, the writer feels that a reasonably high standard of mathematical rigor was maintained in the majority of these courses.

Thus, all the students involved in this investigation had been exposed to a number of courses in which mathematical proof was an integral part of the course. On the other hand, every student had taken at least one course (calculus I I I ) from an instructor who did not make proof a part of his courses. His courses were based on a collection of examples and techniques for problem solving, and he did not prove theorems in class not require proof from the students on assignments and tests.

The usual teaching method of the instructors contacted was lecture in conjunction with varying amounts of class discussion and questions. One instructor reported that he had used "guided discovery 45 fo r a unit of about three weeks' duration in his course.

The performance portion of each study describes the student's progress and reactions throughout and immediately following the real analysis course. This portion is based on information gleaned from the student's diary, classroom reactions, the two interviews, and the instructor's records. Although the comments are necessarily subjective, they are based on a considerable collection of information.

An overall glimpse of the work and achievement of the indi­ vidual students is afforded by table 3 on the following page. The numbers in the columns t it le d Hours Spent Total and Solutions Claimed were taken d ire c tly from the diaries of the students. It is lik e ly that the hours spent was at best an estimate on the part of most students. The solutions claimed refer to all of those that the student felt he had done successfully at the time he did the work.

It is lik e ly that several of these might not have held up in a class presentation. The Original Solutions Presented column gives the number of presentations the student made, for which he was the firs t to present that problem, and there was some merit to his approach.

That is, totally incorrect solutions presented are not included. The numbers in this and the remaining columns come from the writer's indi­ vidual student records. For each original presentation, a score weighted by the writer according to the difficulty of the problem and the quality of the solution was assigned. Although no upper bound was set, in practice only scores from { 1, 2 , 3, 4, 5, 6 } were assigned. Alternate Solutions Presented refers to voluntary solutions given by the students which d iffered in some way from the TABLE 3

WORK AND ACHIEVEMENT

Student Hours Spent Solutions Original Weighted Score A 1ternate Test Overa 11 Claimed Solutions Solutions Score Class Total Ave/Wk.(132 pos) Presented Total Ave/Prob. Presented (100 pos) Rank

1 45 3.2 59 10 15 1.5 1 40 7

2 64 4.6 55 9 20 2.2 0 75 4

3 38 2.7 99 13 41 3-2 4 93 3

4 61 4.4 62 8 12 1.5 1 81 6

5 60 4.3 78 21 50 2.4 5 86 1

6 121 8.7 69 3 4 1.3 2 39 8

7 64 4.6 71 18 39 2.2 5 77 2

8 142 10.1 87 12 16 1.3 1 60 5 original presentation. No attempt was made to weight these. Test

Score refers to the score on the hour test. The Overall Class Rank

is a q u a lita tiv e judgement based prim arily on the student's contribu­

tion to the class.

The remainder of this chapter is devoted to studies of the

individual students.

Student Number One

Background

Student one was a twenty-one year old senior woman with a

dual major in mathematics and English education. Her vocational objective was to teach mathematics in the secondary schools. She was the only student in this investigation who had begun her study of collegiate mathematics with calculus II. This indicates that

she had taken calculus in high school. The early start in calculus made it possible for her to take three post-calculus courses during

her sophomore year. During her junior year, she concentrated on

English courses and took only two mathematics courses, both during

the second semester. Her post-calculus courses included abstract

algebra, linear algebra, probability and statistics, and geometry.

She was a good student receiving one A, fiv e B's and one C in

her mathematics courses. It is interesting to note that the five mathematics courses she took during her freshman and sophomore

years were taught by only two members of the department. Furthermore,

the teaching methods and mathematical philosophies of these two 48 instructors were quite different. Calculus II, calculus III, and probability and s ta tis tic s were a ll presented in terms of problems and solutions with no stress on the theoretical aspects of the subject. On the other hand, in her abstract algebra course, about three weeks were devoted to a unit on group theory during which the students were led to make their own discoveries and provide rigorous proofs of their discoveries.

Student one had not taken any courses from the w riter prior to the real analysis course. Comments from her other instructors indicated that she was f a ir to b etter than average at w riting proofs.

Although not always successful at w ritin g proofs, she appeared to be able to recognize when she was going wrong.

Performance

Student one began the course in good style by presenting several solutions the first few periods. However as the course progressed and the m aterial became more d if f ic u lt , she began to show less achievement although seemingly working. In fact, after the firs t eight class periods, she tended to become merely an observer of the class. Only occasionally did she interject a comment or attempt to present a solution. The presentations she did choose to make were usually quite imprecise although perhaps having the germ of an idea. Her conception of what constituted a proof seemed to be very weak. Near the end of the course, student one again began to make contributions to the class and presented several of the metric space problems. Solutions or attempts at solution for the following problems were presented: 1. 1iii, l.lviii, 1. 5, 1. 6 iv,

1.7 i , 1.7 i i » 1.7vii, 1.12111, 1.14, 1.54, 1.59i, 2.1, 2.3, 2.71-

It appears then that student one was able to handle the simpler problems, but was frustrated by the more sophisticated mat­ e r ia l. She seemed unable to achieve when the proof demanded more than a nearly mechanical application of the axioms, d efin itio n s, or previous problems.

Student one expressed mixed feelings about the course at the interviews. On the one hand, she stated that she liked the course, especially the informal nature of the classes. She also expressed pleasure at the individualized aspects of the course and the fact that everyone had an opportunity to work things their own way. On the other hand, at the midterm interview she seemed to be quite frustrated with her inability to write proofs. The failure to experience a feeling of reward led her frequently to feel that her work was fru itle s s . Feelings of fru stratio n were less evident at the final interview.

Discussion of her difficulty with writing proofs revealed that she tended to blame her mathematical background. She commented on this in her diary essay October 19:

I'm constantly trying to analyse my mind to figure out why I canft see through a proof. I think that it has something to do with my f ir s t college math courses with Dr. Spicer who put absolutely no emphasis on theory. He trained his calculus students to follow formulas without an actual understanding of the theory behind them.

It should be kept in mind that each student involved in this investigation had at least one such calculus course. She further

revealed in her fin a l interview, that in other previous courses

she had been able to rely on memory for proofs:

I had never been required to write proofs before. In other courses I had seen proofs, but was not required to do them. The teacher did a ll the proofs and I ju st memorized them. This was the f i r s t course that I re ally had to prove something.

Although these factors possibly influenced her, it was the writer's

impression that she had little real desire to write proofs. In

fa c t, i t seemed at times that she had convinced herself that she

could not write proofs, so she really could not.

The emotions f e l t by student one as she struggled through the

course are well documented in her diary essays (see Appendix F ) .

Her overall feeling toward the course is summarized in her December 16

essay: "I can honestly say that I enjoyed the course although my

achievement was not what I had expected in the beginning or what I would have liked it to be."

Despite her lack of achievement at times, student one considered

the course a personal success. She made these comments at the fin al

interview:

Even though I d id n 't learn as much as I wanted to or think I should have, I think it [the course] was a success. I think i t changed my attitudes toward mathematics and towards proof. I think they were good changes.

It seems then that this teaching method has affected the

student's concept of the nature of mathematics and her approach

toward learning mathematics. In particular, she seemed to find

that proof and the doing of proofs was a creative process rather

than something that could be memorized. However, based on her 51 accomplishments in the course, i t seems that, at best, she made only small gains in her ability to create her own proofs.

Student Number Two

Background

Student two was a twenty-one year old senior man. He planned to spend two years in military service after graduation, and then do graduate work in mathematics. He began his study of collegiate mathe­ matics with analytic geometry. This indicates an average high school mathematics preparation. He had an average record in mathematics during his f ir s t three semesters of college; however, his performance

in other areas was poor and he was placed on academic suspension during the second semester of 1967~68. A fter returning to college

the summer of 1968, his record in a ll areas was b etter. In p a rtic u la r,

he compiled a good record in his mathematics courses which included geometry, abstract algebra, linear algebra, and differential equations.

His grades in mathematics courses were one A, fiv e B's, and two C's.

These courses were taken from six d iffe re n t instructors including the writer from whom he took abstract algebra. Also, he was writing his

senior thesis (begun during the second semester of the junior year)

under the writer's direction.

In the abstract algebra course, student two seemed to have

unusual insight in following proofs presented in lecture, suggesting

steps in the proofs, and raising questions. Although fair to good

at writing proofs, he usually talked better than he wrote. Comments

from his other instructors seemed to indicate much the same behavior; that is, he was very good at speaking up in class concerning proofs but sometimes less successful in putting his thoughts on paper.

Performance

Student two proved to be capable if somewhat erratic in his performance during the course. At times he seemed to be somewhat reluctant to present his solutions, but was usually correct when he did so. The total number of solutions he presented was small, but they were distributed over the entire semester. On two occasions he presented problems that no other class member had been able to solve several periods a fte r they were o rig in a lly brought up. Thus, although not one of the more prolific students in sheer number of solutions presented, he certainly exhibited the ability to achieve when given enough time. Solutions or attempts at solution for the following problems were presented: 1 .1 iv, 1.3 i , 1.3v, l.lO v , 1.13 i i > 1 .1 3 iii*

1.16, 1,25, 1-38, l.M , 1.59iv.

Student two was on occasion good at offering comments con­ cerning problems or th e ir solutions, a ch aracteristic that he had exhibited in previous courses. In fact, he was described by another student in the course as being a "professional skeptic." He often showed intuitive insights into the problems, although his work some­ times seemed to end there as he failed to exploit the ideas fully.

Thus he frequently appeared to be on the verge of a solution but did not see it completely until someone else presented it.

In the interviews and his diary, student two expressed a certain frustration with the course--especially when his ratio of success to fa ilu re was low. However, he stated that he enjoyed the

course and liked the teaching method. He fe lt that in this course he was able to see why or how things happen, whereas in a lecturehe would

see the professor do a proof and say "okay, I believe it" but would

not re a lly understand why or how i t works or where it came from. Com­ menting further on this at the fin a l interview, he stated:

I think everyone should be exposed to the method. I think that they would learn ju s t as much as they would reading i t out of a book. In fa c t, they would learn more— they would learn more about doing proofs and learning to read mathematics.

It seems doubtful that the course had a large effect on the

student's concept of the nature of mathematics or on his approach to

learning mathematics. It did afford him an opportunity to be chal­

lenged by certain problems and have the time to work them out. That

is, when no one else was able to present a problem, he seemed to

relish the idea of working on it until he solved it. At the final

interview, he mentioned that he was s t i l l working on two such prob­

lems even though the course had o f f ic ia lly ended two weeks before.

In short, the performance of student two in this class was consistent with his previous work. He was able to achieve when he really set

his mind to it.

Student Number Three

Background

Student three was a twenty year old junior man. In addition

to his mathematics major, he was taking courses in economics and

planned to do graduate work in business adm inistration. He was the

least experienced mathematically, having only a course in modern algebra to go with his introductory courses. The fact that he began his study of collegiate mathematics with an alytic geometry indicates an average high school mathematics preparation. However, his work in the introductory courses, taken from four different instructors, was consistently excellent. In addition to straight A's, he received the freshman mathematics award— given to the most outstanding student in freshman mathematics courses.

Despite the small number of courses, student three had experienced a wide variety of teaching methods including guided discovery for a unit on group theory in the modern algebra course.

His instructor for that course rated him as one of the best in the class at writing proof. He had taken no previous course from the writer, although he took abstract algebra concurrently with the real analysis.

Performance

Student three was probably the best mathematician in the class. That is, he had exceptional insights into the problems plus the ability to translate these insights into precise rigorous proofs.

It did not seem to matter whether the problem was easy or d if f ic u lt ; he consistently wrote the best proofs in the class when he chose to participate. His proofs showed a great deal of creativity or unusual approaches. For example, he gave an indirect proof of the Swartz

inequality. He was also good at constructing examples or counter­ examples on the spot and finding errors in proofs. On the one occasion that he presented an erroneous proof, he spotted his own 55 error later in the period and rectified it the following class period.

Solutions for the following problems were presented: l.lii, 1.3i »

1.8M, l.lO iii, 1.21 , 1.28, 1.32, 1 .35, 1.40, 1.46, 1.59, 1.51, 1-53,

1.56, 2.4, 2.7i i i , 2.8.

Despite his generally excellent work, student three was erratic. That is, there would be several periods in a row during which he would have no solutions. The w rite r usually got the impres­ sion that the student was not working during these times or that he just did not care enough to stick with a problem u n til he got a solu­ tion. This opinion was partially verified by the student. During the midterm interview he commented that under the format of the course

it was easy to let the work slide. Examination of his diary col­

laborates this to a certain extent, although there seem to have been some times during which he was working but not achieving. It seems then that even the best students faced fru s tra tio n at times and were unable to do a ll the problems. This fru stratio n was expressed by student three on October 9 when he wrote in a diary essay:

You could say I'm frustrated at this time. It seems that every time I sit down to work analysis problems, I get nowhere. You feel you have wasted time when you scratch for an hour and accomplish nothing.

Several weeks later he seemed to view this frustration with less disappointment. The following remarks w ritten on November 11 seem

to provide some evidence of the success of the Moore method:

It seems that the more I see of the course, the better I like it. The class is relaxed and informal, yet at the same time I feel I'm learning a lot. I find that I get much more out of a proof that I work on for 40 or 70 minutes than one that is thrown at me cold in class, as is usually the case in a course 56

like abstract algebra. Even if you work for an hour and don't prove anything, you get to see the complications and better appreciate the proof when you finally see it presented. This hour of " fru itle s s " work might also enable one to see a loop­ hole in another's proof.

Student three seemed to become especially conscious of the power of indirect proof. Besides the indirect proof of the Swartz inequality, he frequently used this technique in other proofs

(including proofs in the abstract algebra course). Thus there was a visible gain in this aspect of writing proofs. Other than this i t is d if f ic u lt to see any effects the course had on his concept of the nature of mathematics or his approach to learning mathematics. His primary interest in mathematics seemed to be its use as a tool. Although quite capable of doing creative proofs, in the final analysis, he seemed to have little enthusiasm for this aspect of mathematics.

Student Number Four

Background

Student four was a twenty-one year old senior man. He planned to do graduate work in statistics or operations research. The fact that he began his study of co llegiate mathematics with analytic geo­ metry indicates an average high school mathematics preparation. He had compiled a good to excellent record gradewise in a number of substantial mathematics courses which included geometry, abstract algebra, probability, statistics, and differential equations. His mathematics courses were taken from six d iffe re n t instructors, includ­ ing the writer from whom he took abstract algebra. He was enrolled 5 7 in a second abstract algebra course given by the w riter concurrently with the real analysis. In the writer's previous abstract algebra course, he was a good student; however, he had d iffic u lty w riting proofs. This weakness in w ritin g proofs seemed to have been unique to the writer's course since the student's other instructors rated him as good at writing proof. In fact, his probability instructor described him as the only one in the class who consistently knew what was going on.

Performance

Student four did not perform as well as was expected in the real analysis course. Although he was occasionally able to make constructive comments or find errors in the solutions given by other students, he was seldom able to solve the problems himself. Even when he had a solution, he seemed to be reluctant to present it in class. He just did not seem to be able or inclined to operate suc­ cessfully under the format of the course. He presented solutions or attempts at solution to the following problems: 1.3ii» 1.8i , 1.8i i ,

1.15, 1.19, 1,34, 1.4311, 1.55.

The feelings of student four are expressed in this diary essay w ritten November 12:

I am not having success la te ly in proving problems and feel somewhat discouraged. My creativity is not very great and I can't seem to develop it either. At the moment the structure does not seem to help my learning too much. I now see i t is much more d if f ic u lt to formulate a proof than it appears in any textbook. Creativity in developing proofs certainly appears to be at the heart of this mathematics course. Many of these proofs look nice and somewhat easy once you see how to work them. This course shows that d e fin ite ly there is more to being 58

a mathematician than ju st being able to apply math. The basic types of proofs being used I understand; the problem is picking the rig h t one.

In spite of the inability to present solutions, it seems that student four did learn some real analysis in the course—at least he did if the hour test gives any measure of knowledge of real analysis.

On this test he scored eighty-one percent which was third highest in the class. Also for the one problem on the test requiring a proof, he wrote a very precise rigorous proof. This seems to indi­ cate that the student was learning, but just not contributing to the class.

It is interesting to note that in the writer's abstract algebra course during the same semester, student four was one of the top three students in the class. (The other two were among the top three in the real analysis course also .) This course was taught in a more conven­ tional manner with a textbook. The students were required to do a large number of proofs in this course, but they were handed in rather than presented before the class. Contrary to the writer's evaluation in the background portion of this case study, student four was quite successful this time in writing proofs under these circumstances.

Interviews with the student revealed dissatisfaction with the teaching method. He tended to blame his in a b ility to w rite proofs on lack of experience in his previous courses. He also felt that use of a textbook as a reference for proofs should be allowed.

The interviews also revealed that the student did not feel that writing proofs was sufficient motivation for studying the material. 5 9

He suggested that:

Once in a while the instructor should show uses for the m aterial. Explain the relevance of the problems other than just to prove other problems. There is need for application of the material to other areas. For example, how could it be used to solve prob­ lems encountered by an engineer?

Student four was the only student to raise the question of applications at the midterm interviews. Other students mentioned it at the fin a l interviews, but d id n 't seem to be as concerned as student four.

This concern for applications by student four seems somewhat paradoxical in lig h t of his performance in the abstract algebra course.

It seems more likely that his lack of success was due to his feelings about the teaching method. That is, he was just not the type of person who would assert himself to the extent demanded by the method.

At the final interview, student four seemed to be more dis­ appointed with the course than any of the other students. He was not b itte r about i t , but ju s t seemed to w rite the course o ff. His concept of the nature of mathematics seemed relatively unaffected by the course.

He had seen a d iffe re n t face of mathematics in the course, but seemed largely to reject it. He seemed less confident or less interested

in writing proofs than he had before the course. He stated that he had trie d not to le t the course a ffe c t his attitu d e towards learning mathematics. In short, the Moore method seems to have not been a suc­ cessful teaching method for this particular student. Student Number Five

Background

Student fiv e was a twenty year old senior man. He planned

to do graduate work in mathematics, then enter college teaching or

industry. He began his study of co llegiate mathematics with calculus I.

This indicates an above average high school preparation in mathematics.

He had compiled an excellent record in his co lleg iate mathematics

courses which included geometry, abstract algebra, lin ear algebra,

topology, differential equations, and complex analysis. He received

an A in a ll but one of these courses; the one being topology where

no A's were given and his grade was one of two B's. His background

for this investigation was unique in that he was-the only student who had taken topology or complex analysis— two courses with some

content closely related to that of this real analysis course. In

each case he had been the only junior in the class which otherwise

consisted of seniors.

Student fiv e had taken courses from six d iffe re n t mathematics

instructors including the writer from whom he took abstract algebra

and complex analysis. Also he was w riting his senior thesis under the

writer's direction and took a second abstract algebra course from the

writer concurrently with the real analysis. He had been very good at

w riting proofs in his prior courses from the w rite r. He always knew

when he had a proof, and if he had d iffic u lty he would stay with the

problem u n til he worked i t out. Comments from his other instructors

indicated that he was good at writing proof in their courses also. Performance

Student five probably contributed more to the course than any other student. He seemed to relish the give and take of the class sessions especially in the second half of the semester. At times when the classes seemed to drag, he was willing to present his ideas for the solution of a problem or to comment on another student's solution. Although his presentations were sometimes ragged or even wrong, his erroneous e ffo rts frequently paved the way for future correct solutions by him or some other student.

This is not to say that the majority of his presentations were erroneous. In fact, his solutions were usually correct, precise, and rigorous even when they included the most d ifficult problems of the course. Solutions or attempts at solution to the following problems were presented: l . l i i , 1.3 i , 1 .6 v i, 1 .7v i i , 1. 13><>,

1.13 • v, 1.16, 1.24, 1.27, 1.32, 1.39, 1.41 , 1.42, 1.431M, 1.47,

1.53, 1-57, 1.59ii, 1.60, 1.64, 1.67, 2.1, 2.6, 2.10iii, 2.11.

As were the other members of the class, student five was sometimes fru strated . However, his fru s tra tio n seemed to be not only with his inability to prove some of the problems, but also with the pace of the course and the participation of the other class members. This frustration was especially evident near the middle of the course when the class was struggling. On October 29, after 21 class periods, he wrote in his diary:

There are aspects of this method of running the course that bother me. It seems as if we stagnate quite a bit in class. For example in one recent class we worked on one problem and only got part of a solution. I re alize that the purpose of 62

the course is not merely to crank out results. I f this were a ll we wanted we could do ju s t as well by going to a lecture format.

At any ra te , I think we should be covering more analysis than we have been covering. What I'm getting at is that I question whether the results we obtain actully warrant the amount of time that is invested.

Later in this same essay, he commented on the reasons fo r his some­ times ragged presentations.

Another somewhat unrelated point concerns me. This is the amount of class p a rticip a tio n . At times the class drags on with none able to solve a p a rtic u la r problem. In this s itu a tio n , many times I ' l l volunteer to work on a problem even though I have only a partial solution or have gained some insight a few minutes before. Sometimes the results are wrong because in my e ffo rt to f i l l in the gaps I overlook a minor detail. Sometimes I'm not sure if I should do this as much or i f I should hold back on the problem until my results were more polished.

By the end of the course, student five seemed to have a brighter perspective of the course. He had become quite confident of his abil­ ity to write proofs. Also, he apparently felt a sense of accomplish­ ment that outweighed the frustrations. He now seemed to view these struggles in a different light. On December 11, he wrote:

Now that this course is almost finished I feel a real sense of accomplishment in relatio n to i t . You proposed two objectives in the introduction to the course. First, you expected us to do some mathematics. I've never had a course of this type before and I fe e l now that more of my courses should have been done this way. . . . I feel that the involvement here by actually proving theorems myself has helped me to learn much more than I would have in a lecture course covering the same material. Certainly, we did not cover a vast quantity of material but the methods we used to attack this material should make us more able to go out and learn on our own without any help from a professor.

This brings me to your second objective, that of learning some real analysis. What I said above will apply to this objective also. We may not have learned a large number of "facts" about analysis but we have gotten into analysis ourselves and viewed it from within. For these reasons, what we have done has been much more 63

valuable to me than s ittin g in lectures and reciting ideas that are not developed by us but given to us by a professor. There is a need for lecture courses, don't get me wrong, but this course has been extremely valuable. For these reasons I feel that this course in real analysis is probably one of the best that I have ever taken.

This theme was reiterated by the student in his final interview

(see Appendix G); in fa c t, perhaps he overreacted. To use a possibly overused phrase, it seems that student five was "turned on" by this teaching method. Perhaps the teaching method was more successful for this particular student than any other in the class.

It seems then that this course did a ffect the student's concept of the nature of mathematics and his approach to learning mathematics. The changes in his concept of the nature of mathematics were not major--he had done mathematics before. They might even be more in the nature of reinforcement. S till he seemed to become more aware of the power of proof in developing mathematics which he seemed to view as a creative process. He also seemed to rely more on himself to learn. Possibly as a result of this, he became critical of the teaching methods used in other courses, especially the giving of tests which depended prim arily on memory. This seems to indicate an e ffe c t on his approach to learning mathematics which might produce secondary effects in others— most p a rtic u la rly , his teachers. Student Number Six

Background

Student six was a twenty year old senior man. He planned to work in his family's firm as an accountant after graduation from col­ lege. He had entered college under the Summer-February program— a plan whereby students who did not meet the normal minimum academic standards for admission were permitted to come to a summer session, and then were admitted to the college the following February for the second semester. Student s ix 's f i r s t collegiate mathematics course was a review of algebra and trigonometry. This further indicates below average high school mathematics preparation. In addition to introductory mathematics courses, he had taken courses in geometry, abstract algebra, linear algebra, probabi1ity, and statistics, but his record was mediocre with two B's, four C's, and fiv e D's in mathematics courses. He had taken mathematics courses from seven d iffe re n t instructors including the w rite r from whom he took abstract algebra. In the abstract algebra course, he seemed to have no concept of what a proof was. His attempts at proof had usually seemed unrelated to the statement he was trying to prove. Comments from his other instructors seemed to agree with the writer's evaluation.

Performance

Student six was able and willing to present problems during the f ir s t few class sessions. His solutions during this stage were generally correct. However as the material grew more difficult, he 65 soon appeared to be out of i t . In fa c t, a fte r the f ir s t three weeks, i t was a rare day indeed when he did more than come to class, s it q u ie tly , and watch the other students present th e ir work. He was the only student in the class to do one of the optional hand-in assign­ ments. He presented solutions or attempts at solution to problems: l.lii, l.liii, 1,lvi i i, 1, 311, 1.6iv, 1.121, and 1.13v.

On the rare occasion that student six made a constructive com­ ment in class, he sometimes showed an understanding that came as a surprise to the writer. For example, this comment was taken from the individual student record for class thirty-six:

Surprisingly enough [student six] can make comments that indicate an understanding of the material. He apparently just doesn't do any work. When asked .about problem 2 .1 , he had no idea what a metric space was. However, a fte r looking at the d e fin itio n while examples were being given, he was the f ir s t student to see that the example, d(x,y) = |x2 - y2|, x,y e JR, would not satisfy part ii o f the defini tion.

The w rite r's comment about student six doing no work was pos­ sibly unjustified. The student indicated in his diary a considerable amount of time spent working throughout the course. However, his diary reveals that he had not yet begun studying the metric space problems at the time this comment was w ritte n . As a matter of fact, his diary seemed to indicate that he seldom worked on the problems more than a day or two in advance. It also seemed to indicate that especially in the latter stages of the course, he didn't look at the problems u n til a fte r they had been discussed in class. This seems to indicate that he was studying similarly to the way one might in a lecture course—that is, copy down the proofs and then try to figure them out la te r. Thus this teaching method seems to have had l i t t l e 66 effect on his approach to learning mathematics.

In spite of his apparent lack of achievement, student six seemed to maintain a positive attitude toward the course. He wrote very little in his diary essays, but in the comments he did make, he almost seemed to view his lack of achievement with detachment. This was in contrast to some of the other students who seemed compelled to express their frustrations at times. On November 11, student six wrote: "The proofs I have are not as detailed as they should be. I am having more difficulty as the course goes along." Near the end of the course, he wrote:

In the first part of the semester, I contributed in class pretty well, although in the last part I did not. This is not to say, however, that an effort was not put forth. I wasn't confident in much of the work I did and did not feel that my proofs were of as much value as those of the rest of the class.

In response to questions during the interviews, the student indicated that he liked the course and the manner in which i t was taught. He said that he f e l t motivated to work under this format where he could set his own limits. He commented several times on the informal nature of the course, stating at one time: "I feel a bit freer to stick my neck out." Continuing he said: "In the informal class, there is more room for comments from others. There seems to be more eagerness to understand." The only suggestion for change he offered was that more examples should be included.

The comments made by this student seem almost paradoxical

in lig h t of his achievement. However, as examination of his back­ ground shows, fa ilu re to succeed was not a new thing fo r him. In 67 this case the Moore method was probably no b etter or no worse than any other method.

There seems to be little evidence that this teaching method affected the student's concept of the nature of mathematics. In fa c t, he seemed to not really understand very much about mathematics. For instance at the final interview when questioned about the role of axioms, theorems and proof in mathematics, he replied:

Axioms are minor proofs in themselves. You distinguish axioms from proofs in that proofs are more general and the axioms are evolved from the proofs—are certain work-outs of the larger proofs and the larger theorems.

Case closed.

Student Number Seven

Background

Student seven was a twenty seven-year old senior man. He planned to do graduate work in computer science. He came to Western

Maryland College at the beginning of the second semester of the 1967-68 academic year a fte r unsuccessful e a r lie r attempts at college and a stint in the U. S. Navy. He immediately began to distinguish him­ self in his mathematics courses and compiled an impressive record of straig h t A's in mathematics courses which included modern algebra, geometry, probability, and statistics. In addition to his success in coursework, he was a successful assistant for both the mathematics department and the computer center.

Student seven had taken mathematics courses from fiv e d if f e r ­ ent instructors. In his modern algebra course, he had been exposed to a unit on group theory which was taught by guided discovery. 68

Comments from his instructors indicated that he was one of the best at writing proofs. Although the writer had not had student seven in class, he had had a number of wide ranging informal discussions with him concerning various areas of mathematics. The writer was partic­ ularly impressed in these discussions with the student's ability to understand unfam iliar mathematical ideas and his willingness to work

independently on problems which arose during these discussions. In addition to these informal activities, the student was working on his senior thesis under the writer's direction.

Performance

Student seven was a consistent worker in the course. He probably struggled more with the material than student three or student fiv e , but his achievements were on a par with th e irs . Also,

it was fe lt that student seven gained more in mathematical maturity

than any other student in the course. Solutions or attempts at solution to the following problems were presented: 1.2, 1.61, 1.91,

1.11, 1.1311, >.18, 1.23, 1.31, >.37, 1-39, 1.42, 1.431, 1.45, 1.48,

1.49,1.53, 1.56, 1.57, 1.59111, 1.65, 1.66, 2.7iv, 2.9.

In the early part of the course, the proofs of student seven

tended to be distinguished by d irec t attacks on the problem. That

Is, he seemed to apply brute force to push the solution through.

Although generally successful, this occasionally led to oversights

and failure to take into account special cases. A spectacular example of this was his attempt at proving the w ell-ordering prin­

c ip le , problem 1.18. He claimed a solution to this problem by using 69 mathematical induction on the number of elements in the sets. That

is, his inductive hypothesis was that every set of positive integers with n elements has a least element. ( I t should be noted that the

rest of the class also failed to spot a fallacy in this proof. Further­ more after some skepticism evidenced by the instructor and to the

credit of student seven, he noticed that the same proof could be

used to show that every set of positive integers has a greatest

element. This led to a very good class discussion on the limitations

of mathematical induction.)

As the proofs became more complicated, this direct attack

perhaps produced fewer successes. Student seven seemed awed by

some of the presentations. At the midterm interview, he commented:

In some of these proofs all the little pieces seem to fit together— it is like a little chain up on the board, this one goes to that one and so on— but you stand back and look at i t and it doesn't seem possible to go from the first to the last. The l i t t l e pieces f i t together, but the whole thing doesn't.

Trouble with the proof of statements that involved infinite sets

also plagued the student through much of the course. He was espe­

c ia lly upset with the proof of problem 1.27 (Heine-Borel) . However,

in spite of th is , he presented a very elegant proof fo r problem 1.31

(Bolzano-Weierstrass) using the Heine-Borel property.

Near the middle of the course, student seven frequently

trie d to ta lk his way through proofs at the board, rather than w rite

out all the details. He seemed to be getting the big picture, but

was reluctant to fill it in. After considerable prodding by the

w rite r and class, his proofs at the board became more complete in the la tte r stages of the course. He also became more aware of the necessity fo r checking the d e ta ils . For example, problem 1.65 is a false statement. Student seven attempted to prove it and saw that on the surface at least everything seemed to go through. However, in checking the details, he could not quite get it to fit together.

Thus, he came in several days before i t would come up in class and asked the instructor to look over his proof. He pointed out where he was having difficulty. The instructor agreed that there were some d iffic u ltie s there and suggested that he consider the problem some more. During the next few days, the student commented that he was s t i l l having trouble. He was given encouragement, but no further hints. When problem 1.65 came up in class, student seven was called on to present i t . Much to the w rite r's surprise, he proceeded to produce a counterexample to i t . It is doubtful that he would have done this e a r lie r. (There were occasional relapses however. Student seven immediately proceeded to present a proof of the fa lse statement in problem 1.66.)

Student seven seemed to have mixed emotions about the course.

At the midterm interview, he stated:

I enjoy the challenge but feel much frustration. Frustration is part of a challenge though—you have to be frustrated before a challenge is worthwhile. Maybe it is just like an interim period—maybe things will begin to look up and I wi11 really understand things better because I have had to struggle with them.

The following excerpt from his diary essay w ritten October 15 reveals an attempt to analyze the reasons for his occasional lack of success and the resulting frustration. 71

Possibly my problem, or one of my problems is a lack of expe­ rience and being accustomed to having a proof laid out in front of me. Possibly it is always having been able to use direct methods to achieve the proof. . . . Possibly if I analyze a given problem f i r s t , re fle c t on the methods we've used for proof and don't get hung up on brute force proofs, my luck with the problems w ill improve.

Another area of concern fo r student seven was the amount of m aterial the course covered. As early as September 26, a fte r seven class meetings, the student questioned whether he would learn as much real analysis under this teaching method as the students had learned the previous year. He raised the point again during the midterm interview and reflected on it in his last diary essay written

December 18:

Looking back on the course I wonder whether we learned as much real analysis as people would in a course with a traditional structure. Possibly we've learned, or have been guided in the direction of learning something more important than statement of theorems. This something is the a b ility to work with mathe­ matical concepts through the form of proof.

Despite his reservations, this teaching method seems to have worked well with student seven. His understanding of proofs was certainly strengthened. Whether the teaching method affected his concept of mathematics or his approach to learning mathematics is questionable. He stated at the final interview that it did not change his view of mathematics--he already knew that mathematics was more than calculus. He was also used to learning mathematics on his own before this course. Still it seems likely in light of his growth in the course that he must have f e lt an increased aware­ ness of the role of proof in mathematics. Student Number Eight

Background

Student eight was a twenty-one year old senior man. He planned

to enter military serivce upon graduation, then do further college work in computer science or operations research. He had entered college under the Summer-February program described e a rlie r. He began his study of co lleg iate mathematics with a course which reviewed algebra and trigonometry. This indicates below average high school mathematics preparation. In addition to introductory mathematics courses, he had taken courses in abstract algebra, linear algebra, probability, and geometry. His college record prior to his senior year was inconsistent. In spite of poor grades in mathematics courses--three C's and six D's—his overall performance in college had been satisfacto ry. This is perhaps partly due to his interest

in another area in which he took a number of courses although not

declaring it his major.

Student eight had taken courses from seven d iffe re n t mathe­ matics instructors including the w rite r from whom he took abstract

algebra. In the writer's course, he had seemed to have no concept of what proof was about. This judgement seemed to be borne out by

comments from his other instructors.

Performance

Student eight volunteered for the initial presentation of

the course. He continued with occasional presentations throughout the course. This is somewhat remarkable in light of the fact that after the introductory material, his presentations were sometimes totally incorrect or revealed a definite lack of understanding.

Yet, he seemed motivated to work and willing to share his ideas.

Solutions or attempts at solution to the following problems were presented: l.li, 1.3ii, 1.7M, 1 - 7v i i i , 1.11, 1.13 i i i , 1.20, 1.22,

1.35, 1.40, 1.53, 1 • 59i i i . 1.63.

There were instances at which student eight showed the ability to do mathematics. For example, during class nine on

October 1, he presented a proof fo r problem 1.22: The set JN of a ll positive integers does not have an upper bound. Members of

the class were quick to point out that he had only shown that no positive integer could be an upper bound. He said he would work on

it more for the next class meeting. At the following class meeting, he presented a complete correct proof of the statement. He clearly had seen the deficiency in the initial attempt and corrected it.

The diary of student eightrevealedthat he spent more time on this course than any other student in the class. He commented

several times about the work he was putting into the course at the midterm interview.

I am d e fin ite ly spending more time on this course than in lecture courses— sometimes a ridiculous amount of time. No doubt about i t . In lecture courses I have a tendency to miss assignments. There is not so much pressure to get in there and spend more time. I am working almost everyday on this course because I know that I am going to be called on.

When questioned as to whether the time spent was worthwhile, he replied It is worthwhile i f I prove something; it 's not worthwhile i f I don't prove something. . . . However, even i f working in circles, you learn by trial and error and error [s ic .]. You have a b etter insight into i t when you see somebody do i t and see that it actually can be done.

The time spent evidently achieved, along with the frustration, a modicum of success that enabled him to maintain a positive attitude toward the course. Excerpts from his diary essays reveal these fe e l­ ings. Early in the course on September 27, he wrote:

My in it ia l reaction to this course is mixed. I lik e the course because I'v e realized that we are in the process of building a system— a fundamental part of mathematics. Opposed is the b itte r frustration involved— I guess that, too is a part of mathematics.

Further into the course, on November 23, he wrote:

I'm getting bogged down I'm afraid. Not that I'm over my head, but just that there seems to be a catch to some of these ques­ tions. . . . I have to work slowly and deliberately in this course, and if I do that then I usually come up with results. I can see what "rig o r" means--at least p a r tia lly . It takes a lot of concentration and detail to evade the loopholes that can be shot into a proof.

And near the end of the course, on December 16, he wrote:

Well, the semester has drawn to a close and I believe I've learned a lot. . . . I won't be taking 404 next semester, upon your advice, but I would be looking forward to other courses that f o l­ lowed the informal style as encountered in 403. I only hope that my grade in this course reflects the amount of e ffo r t that I have put forth from day to day. . . . But no matter what the format on grades, I believe that the course has been worthwhile and recommend its continuance.

Student eight perhaps more than the other members of the class seemed concerned with the grades to be given. He favored a p a s s -fa i1 system rather than letter grades.

At the final interview, student eight talked at length about how much he enjoyed the course and his enthusiasm for this teaching 75 method (see Appendix G). Although his achievement seemed small in

comparison with some of the other students, this teaching method seemed to draw him out to an extent not seen before. (In the writer's abstract algebra course the previous year, the student seemed very

reticent and uninvolved with the course.) His attitude and involve­ ment in the real analysis course then came as a pleasant surprise.

The teaching method seemed to reach this student better than other methods had in the past.

Although not always able to achieve, the student did seem

to gain an understanding about proofs. As he stated in the fin a l

interview:

I think it [the course] affected my understanding of what you have to do to prove something. It gave me insight into rigoi— what you should do to get a rigorous proof. I can 't say that I'm an expert on i t , but I have much b etter insight into what a rigorous proof should have and what you should do to get a rigorous proof.

His approach to learning mathematics also seems to have been

affected. Besides being w illin g to spend a great deal of time in

the real analysis course, he seemed more aware of things in other

courses. Quoting again from the fin a l interview , he said:

I am more conscious of statements that say "it is obvious" or " i t is easily done" and such cheap words in books. The course has made me a lo t more c r itic a l about what I see. I tend to be more skeptical about statements u ntil I see that they have really been proved. 76

Summary

Students at both ends of the achievement spectrum seemed to

like the teaching method. Although certain elements of fru stratio n were evident in the case of each student, in the end this was appar­

ently outweighed by achievement in the case of some of the students

or taken in stride by the others. Seven of the eight students consid­

ered the course to have been a personal success. Only one student

seemed to have a negative a ttitu d e toward the method.

On the basis of their previous records, five of the students

performed much as expected; two performed at a level below th e ir

previous records; and one at a level above his previous record.

Students three, fiv e , and seven stood out from the others in

their classroom performance. In doing so, they maintained the record

of excellence that is evident in th e ir backgrounds. All three showed

maturation in their ability to create proofs and work independently.

The performance of students one and four was somewhat below

what might have been expected. However, student one seemed to enjoy

the teaching method despite her lack o f success. Perhaps for the

f i r s t time she saw proof in mathematics as something to be done rather

than memorized. Student four a ttrib u ted much of his lack of achieve­

ment to the teaching method. Although his classroom performance was

slight, his test score seemed to indicate that he had learned some

real analysis.

The performance of students two and six, although far apart

on the achievement scale, was consistent with their previous records. 7 7

Student two was a b etter c r it ic than performer. S t i l l he took advan­ tage of lapses by the top three which afforded him an opportunity to do creative work. In these instances, he did work on a par with the best and perhaps at a level he had not achieved before. Student six achieved very little and seemed to have gained little in his under­ standing of mathematics. In light of his previous record, this was not e n tire ly unexpected.

The achievement of student eight was s lig h t in comparison with some of the other members of the class, but was at a level above his usual performance. He seemed to be motivated by the teaching method.

O verall, then, the method in some sense seemed to be suc­ cessful with each of these students. Their individual performances, though varied, were generally consistent with th e ir performances in other courses taught by other methods. The variety in achievement seemed to be no greater in this class than that of previous real analysis classes taught by the writer.

Certain questions concerning the Moore method are examined in lig h t of these student performances in the following chapter. CHAPTER IV

EXAMINATION OF QUESTIONS

In Chapter I , several questions were stated concerning the

Moore method and its relationship to the teaching of undergraduate mathematics. These questions will be considered in this chapter in

light of the discussion in Chapter II and Chapter III. It should be kept in mind than any conclusions drawn are with respect to the

teacher, course, and students involved in this study.

The Questions

1. Is it necessary that the material be relatively unfamiliar to the students? In particular, what happens when the initial m aterial is something as fa m ilia r as the real numbers?

At the initial class meeting, it was explained to the students

that much of the material, especially at the beginning of the course, would be fa m ilia r to them. I t was further explained that they were

building a mathematical system from the axioms and as a result of

this, their proofs must rely on the axioms, definitions, or previously

proved statements. As noted previously, there was skepticism as to

whether a ll of the properties could be proved. However, this skep­

ticism soon disappeared when the proofs were forthcoming.

Actually the students seemed fascinated by the fact that they

were proving things they had assumed previously. They also liked the

78 idea of putting this foundation under th e ir la te r work. The follow ­

ing excerpt from a student's diary is believed to be typical of the fe e li ngs.

The f ir s t 13 problems provide us with a chance to actually see that a ll the properties of numbers could be proved, not ju s t believed to be true. The format of the course seems to lend itself well to the material so far.

This early material provides a stronger foundation in showing us exactly why all of these properties known for so long are true.

The fact that the students had proved many of the properties of real numbers in th e ir modern algebra course was sometimes evident.

That is , a student would state that he remembered proving such a property before. Thus he was able to reconstruct the proof partly from memory. Further into the course, there were times at which proofs were p a rtia lly recalled from calculus, or from topology in the case of one student. However, the frequency of these occurences was not at a level considered to be detrimental to the course. At

the final interview, the students generally felt that prior knowledge was of l i t t l e , i f any, value in constructing th e ir proofs. Indeed some of the hints given were probably more e x p lic it than what a student would recall from another course.

In short, the writer feels that as long as the appropriate explanations are given for studying the material, there is no need

for the material to be logically primitive or isolated (that is,

unfamiliar) from that of other courses. 2. How does an average class (that is , a class in which the students are not selected prior to the course) respond to this method?

With the exception of the junior student who was invited to

take the course, the students in this course enrolled in it as an

elective. As an examination of their backgrounds revealed, the

previous achievement of the students was quite varied. The class

seemed to contain a representative sample of the senior mathematics majors.

In many classes in the writer's experience, there have been

certain students for whom it was felt that the course was a failure.

These students generally tended to give up at some stage of the

course, quit working and generally express bad feelings about the

course. In the experimental course, there was no evidence of a

single student giving up completely. That is, although at least

one student seems to have given up on making presentations in class,

he continued to work on the problems. This was surprising in light

of the previous performance of some of the class members. To be

sure there were occasional feelings of defeatism, a great deal of

frustration at times, and considerable lack of achievement on the

part of some students. However, through i t a ll the students con­

tinued towork and maintain a positive attitude.

The "star" students described by Wilder were certainly

evident in this course. However, i t is questionable whether the

stars profited from performing before the non-active portion of

the class. The stars frequently tended to complain about the pace

of the course and the lack of participation by the other students. At the final interview, the stars recommended selection of the class members as a means of improving the course. On the other hand, it was f e l t that the non-stars profited from the presence of the stars.

That is, they seemed more free to criticize and question their peers

than the instructor.

3. Is the Moore method adaptable to exclusively undergraduate classes? In particular, can undergraduates be sufficiently motivated to study mathematics in this manner? And if so, how much intuitive material must be provided by the instructor?

It seems evident that for this class, the answer to the first

two questions is yes. The students generally liked the method and seemed for the most part motivated to study. However, the role of

the instructor in providing motivation was greater than anticipated.

As previously stated, a considerable amount of interplay between the class members and instructor was a part of the course. This frequently

resulted in a large amount of talking, albeit informally, by the

instructor during some of the class periods. Also periodic overviews of the upcoming material were given by the instructor. This was

intended to provide motivation by letting the students in on where we were going or what the material was getting at.

A certain amount of intuitive material was built into the

course notes. Beyond this very little in the way of intuitive material was provided by the instructor. This lack of intuitive m aterial was a frequent source of comment from the students. They

generally seemed to feel that there was a need for more illustrations

and examples other than those they were able to provide for themselves. Various ways of providing this material were suggested by the students: include more numerical examples in the course notes, draw out examples and illustrations from the students in class discussions, present short supplementary lectures during class. k. How w ill a student's concept of mathematics and his motivation for studying it be affected by a course taught in this manner?

For some students there seemed to be no effects. This

reaction occurred at both ends of the achievement scale. Overall though, the students seemed to be quite conscious of the building of a mathematical system. There was some fascination in this which possibly resulted in an increased willingness to work because they felt it was their own creation. They seemed to feel that they were

inside the system now, doing i t themselves rather than outside watching.

These attitudes were probably more evident in the less successful

students. The better students seemed less affected perhaps due

to th e ir better developed study habits.

5. Can the Moore method be successfully adapted by an average teacher of college mathematics? Or, is i t the case, as Moise suggests, that ". . . the method may require that the teacher be a genius"? [29: 409].

The method certain ly does make strong demands on the teacher.

The intimate involvement with the class requires considerable caution.

In a situation such as this where the teacher talks re la tiv e ly l i t t l e ,

there seems to be more attention paid to any remarks that he does

make. Thus the hints or comments he makes go fa r toward encouraging

the students to keep working, or on the other hand, k illin g incentive. A well-intentioned but misplaced remark may have effects out of pro­ portion to its significance.

It does seem that a certain confidence in one's own ability is needed. The class sessions are quite open-ended, and the ques­ tions that can result from the intellectual stimulation of the class are sometimes challenging. As a student commented at the midterm interview:

I think it [the teaching method] is a more difficult technique than straig h t lecture, because this way you are subject to more questions. We have more time to think about the m aterial and bring up questions.

S till, although being a genius might help, it may not be necessary. At least in the case of the writer (who does not claim to be a genius), the method seems to have been successful. This is not to say that the method could be used successfully by every col­ lege teacher. It is the writer's suspicion, however, that adaptations of the method could be used successfully by a great many teachers.

6. What effect does this teaching method have on the creativity of undergraduate students?

It seems likely that the variety of proofs and techniques for proof that the student attempts and is exposed to must certainly stim ulate his c re a tiv ity . The extent to which the student takes advantage of this stimulation to develop his creativity seems to be tied to his success in the course. That is, the more successful students tend to keep searching for new methods and techniques as they try new angles of approach on d if f ic u lt problems. They seem to accept the challenge to extend their knowledge and at the same time develop th e ir creative powers.

Summary

Six questions initially raised in Chapter I concerning use of the Moore method have been examined in th is chapter. Although d e fin itiv e answers were not given, comments concerning the method in light of the teacher, course, and students involved in this study were presented.

Limitations, conclusions, and further points are discussed in the following chapter. CHAPTER V

INFORMAL CONCLUSIONS

In an exploratory and expository study of this type, there is no s ta tis tic a l "proof" of hypotheses on which to base conclusions or opinions. Rather, conclusions are the writer's interpretations based on examination of the available evidence. In this final chapter limitations on this evidence are given and certain conclusions are stated.

Beyond this, a major portion of this chapter is devoted to discussion of impressions gained by the writer during the progress of this study. These impressions include examples of evidence on which they are based. However, they are not to be taken as firm conclusions. As befits an exploratory study, a number of questions are included with the impressions. Implicit in some of these ques­ tions are suggestions for further study. Other suggestions for fu rth e r study are included in the concluding remarks.

Limitations and Disclaimers

There are certain things that must be taken into considera­

tion in interpreting the evidence in this study. Among these, the source of information is probably of primary importance. There are also certain factors in addition to the teaching method that could have influenced the student's attitudes and performances. And there

85 86 are extrapolations that could be made but are not intended or claimed by the wri te r.

Data sources

The data for this study consisted primarily of diaries and records kept by the students and the instructor, and the two inter­ views with each student. In addition to th is , there was less formal feedback from a number of sources. Students involved in the study frequently discussed various aspects of the method outside of class with the writer, other faculty members and other students. Thus, the teaching method created a certain amount of interest in the mathematics department among both the faculty and other students majoring in mathematics. In addition to comments from the students involved, then, a number of comments were relayed to the writer in discussions with these other people. The value of this informa­ tion may be questionable, but it did play a small role in the writer's thi nki ng.

Influence of grades

The fact that the students were receiving a grade based on their performance must be taken into account. For two of the stu­ dents, a grade of at least C was almost essential for graduation with their class. Two other students had received A's in all of their previous mathematics courses. Grades, then, were a motivating facto r.

In another vein, it is possible that the fact that they were being graded influenced their comments and reactions in their diaries and interviews. This is considered by the writer to be unlikely or at most to have played only a minor role. For one thing the diaries were collected and the fin al interviews were conducted after the students received their grades. Also the comments at the midterm interview, when th e ir grades had not been determined, were generally more c r itic a l of the course and teaching method than they were at the fin al interview. And perhaps most importantly, the writer feels that he was trusted by the students.

Influence of the study

The students were aware that they were involved in the w rite r's doctoral studies. Since Western Maryland College is not a research-oriented institution, it is likely that this was the first time any of the students had been part of such a study at the college level. Perhaps as a result, there was some curiosity about the study. However, a fte r the in itia l c u rio sity , there seemed to be very l i t t l e concern other than to wish the w rite r well at the end of the semester. It is possible, but considered u n likely, that the students' awareness of th e ir involvement in this study contributed to their behavior or reactions in the course.

Influence of friendship

The friendship between the students and instructor should not be overlooked as a possible factor in th e ir attitu d e toward

the teaching method and their achievement. All of the students 88

seemed to like the instructor. As one of the students who had not

known the writer prior to the course commented: "I was never afraid

of you." Also several of the students had become friends of the writer outside of class prior to the course. These friendships

continued and others developed outside of the classroom during the

semester of the study.

The informal nature

The informal nature of the real analysis class sessions was

not entirely due to the use of the Moore method. Informality seems

to be a characteristic of the writer's classes regardless of the

teaching method. In fa c t, i t was commented on in course evaluations

by students in both of the w rite r's other courses during the same

semester. The teaching method in these two courses, although not

s tr ic t ly lecture, more nearly resembled the lecture method than the

Moore method.

Specificity of the class

The study was based on the performance of eight students in

Mathematics **03 at Western Maryland College during the fa ll semester

of 1969-70 taught by the writer. With different students, a different

course, a different place, a different time, or a different instructor

the results might well have changed considerably. Comparison with lecture

Something more should be said about comparison of the

Moore method and the lecture method. As was stated in the f ir s t chapter, it is not the purpose of this study to claim one method superior to the other. However, making certain comparisons seemed to be unavoidable. These are evident in the students' comments and the comments of the w rite r. Although perhaps reflectin g the w rite r's prejudice at times, these remarks are prim arily due to the contrasting nature of the two methods. I t is not claimed that the Moore method is superior to any other teaching method.

Some Conclusions

Even with all the limitations and disclaimers of the pre­ ceding section, i t seems that the evidence presented in Chapters II and III is sufficient to warrant certain conclusions.

1. The teaching method used by the w rite r was successful

in the sense that the students:

a. liked the course,

b. were challenged by the material,

c. were motivated to attempt creative work rather than

rely on memory,

d. saw a variety of proofs and proof techinques used,

e. were able to work independently,

f. with one exception, considered the course to be a

personal success. 90

2. Students with a previous record of high achievement maintained their position.

3. Students with a previous record of low achievement did at least as well as they had previously, and appeared to be motivated to work.

k. All students had an opportunity to do mathematics as well as see mathematics being done.

5. The writer is enthusiastic about use of the teaching method.

Opinions and Questions

Certain points remain to be considered that are not easily placed in any particular category. Included in these are certain hunches or opinions of the writer that are difficult to justify with concrete evidence. They sometimes lead to questions as well as declarative statements.

The role of the instructor

Although with the Moore method, the in structor's presence in the classroom would not be as obvious to an observer as that of a lecturer, it is nonetheless of primary importance. The quote by

Wilder given earlier puts this so well that it is worth repeating.

"Teaching in this style is not merely a negative matter of not

lecturing; rather, it is a striking example of the art that conceals art, and it makes great demands on both mathematical and psychological depth." [29: A09].

Several statements have been made earlier about the involve­ ment of the instructor with the class. Perhaps the most important phase of the instructor's involvement concerns the hints that he gives or does not give. Some examples of this are given in the fol­

lowing paragraphs.

In the experimental course, the order in which the problems were listed in the course notes was sometimes a h in t. The students soon discovered this and made use of it. (This also involved chang­

ing the order.) However, the w rite r was generally reluctant to give additional hints for the problems. One reason for this reluctance has already been given; that is, the students showed a tendency to just wait for hints when they got stuck on a problem. A more impor­

tant reason was that the hints tended to channel the students'

thinking about the solutions. At times this was deemed necessary

in order to stimulate work on a problem. Hints, then, could lead

to a solution of the type the instructor had in mind. On the other

hand, this directing tended to cut down on the variety of proofs

presented for a given problem, and this diversity was considered

to be an important part of the course. A hint could also lead a

student to abandon an avenue that i f pursued would have led to a

solution d iffe re n t from and perhaps b etter than the in stru cto r's.

Despite the fact that several instances of successful hints

have been given in e a rlie r sections, there were also instances when

hints were not visibly successful. For example, no student was able

to present a complete proof that a Cauchy sequence of real numbers

converges despite these hints given over a period of several days:

Hint 1: Lemma. A Cauchy sequence is bounded. (Proof supplied by instructor.) 92

Hint 2: If the range of the sequence is in fin ite , consider the Bolzano-Weierstrass theorem.

Hint 3*' By the lemma and the Bolzano-Weierstrass theorem, the range of a Cauchy sequence has a cluster point. Prove that this cluster point is the sequential limit point.

Fortunately, there were some original thinkers in the class.

Thus on occasion a student would ignore any hints and proceed in his own manner. For example, to prove the Archimedean p rin cip le, problem 1. 23, the writer would show that its denial led to the statement that the positive integers were bounded above. Thus, in the notes problem 1.22 stated that the positive integers are not bounded above. Proving that the positive integers are not bounded above is usually done in d irec tly and is somewhat tric k y .

I t was expected that the class would have d iffic u lty with problem

1.22 but would see that problem 1.23• the Archimedean p rin c ip le , followed readily from i t and would proceed in that order. Instead, a student saw that problem 1.22 would follow easily from problem

1.23. He thus presented problem 1.22 as a corollary to problem 1.23.

Another student presented 1.23 independently of 1.22, so this approach was allowed. Thus, what is usually a tric ky proof was avoided.

There were also instances when students ignored hints given in class, pursued th e ir own ideas, and obtained interesting results.

For example, the indirect proof of the Swartz inequality was obtained despite the standard trick (Let A = Z|aj |2, B = £|b;|2, and C = Eajbj.

Consider E|Ba; - Cbj|2.) being given in class. Another example of original thinking occurred during the development of the d iffe re n tia tio n formulas. To prove the quotient formula, (£)1 = — 1-iL— (with the appropriate restrictions), it was suggested that it might be easier to prove that (—)' = 9 g first, and then apply the product formula. Instead, a student offered this interesting observation: By the product formula,

(fg) * = f'g + g 'f . Hence, (fg) , - g ,f = f'g => f 1 = — ~ ■ .9—-

„> (fa). = (fai'.a.:..-9l(.f.9>-. Let h - fg . then (H)' = g g2 g g

The personality of the instructor

The w rite r has already commented on his own informal nature.

The low key approach to teaching which results from this informal nature is not something that has been noticeably cu ltivated. It does seem to be appreciated by the students. But, is this type of personality necessary or desirable for successful handling of the

Moore method? More generally, is there one type of personality that is more suited to this method of teaching than others?

It is the writer's hunch that the method could be used successfully by teachers of differing personalities. The idea of the teacher's personality was pursued obliquely with some of the students at the final interview. Their reactions were mixed. Some felt that the writer was probably the only one in the Mathematics

Department at Western Maryland College who could successfully use this teaching method; others said that it would make no difference.

Perhaps the main challenge to the teacher using this method is to his self-discipline. After being used to telling the students things, i t is hard to remain s ile n t and le t them struggle to work out th e ir d iffic u ltie s . This is perhaps in the category of "teaching old dogs new tricks." Would a beginning teacher have fewer of these conditioned responses, and thus find it easier to use this method than one whose experience included teaching by the lecture method?

Proof techniques

Probably the biggest difference in the real analysis course and other courses the students had previously taken was the number of proofs and the variety of proof techniques that they encountered.

The case studies of two students in particular show evidence of the influence different methods of proof had on their achievement. One of the students became very prone to use indirect proof and was very successful at using it. He stated once near the end of the course that everytime he did not see a direct proof right away, he auto­ matically tried an indirect proof. Another student who tried every­ thing directly early in the course began to search for finesse func­ tions later in the course and had a startling experience with counter examples. (Overall the number of problems requiring counterexamples was small. This was perhaps a deficiency in the course notes.)

Although the effects were less evident in the cases of the other students, it seems unlikely that they failed to try proof techniques other than direct proof. A pertinent question seems to be: Do students who have experienced the Moore method tend to use techniques such as indirect proof and counterexample to a greater extent than other students? 95

Independent study and learning

One of the advantages to the students of the Moore method seemed to be the opportunity it afforded them to work on the prob­ lems over a period of time. By having the notes available several days in advance, they were able to work ahead as fa r as they wanted to and to keep returning to problems they were having d iffic u lty with. Students frequently commented that even when they did not get the solution of a problem, the fact that they had worked on it helped them understand it when it was presented in class. The value of this work without v is ib le success is something d if f ic u lt to measure, but the writer feels that this work was of considerable value to the students' learning of real analysis.

Nietling [33] devoted considerable attention to the stimula­ tion of "productive thinking' on the part of the students in his study. He had the students attempt to solve problems unfamiliar to them, but designed to lead them to the discovery of mathematical

ideas. The thinking involved in seeking a solution was considered to be an important part of the process. As Nietling states: ". . . one of the basic assumptions was that the typical student would not be completely successful. Indeed, the goal was student participation

in the examination of a situation which could stimulate student

learning." [33= 21]. The writer's opinion with regard to the stu­ dents' learning of real analysis in this study seems to support

Nlet ling's idea.

Problems that were not immediately solved by a student

produced other effects. At least some o f the students seemed to think about these problems at various times other than when they were working s p e c ific a lly on real analysis. They tended to stop by

the writer's office with comments such as: I just had an idea for

problem while s ittin g in psychology class. Do you think this would help get the solution? One student described this phenomenon at the final interview. He felt that his diary time might not be accurate because:

I didn't carry it with me everytime I started thinking about analysis. I might be down in the g r ill and maybe I would think of something and I would w rite i t down or try i t then. Or, I would be in the lib rary and a ll of a sudden a thought would flash in my mind, or before class I would think o f something. I had my diary in my room and I would f i l l it in when Iworked at night in the room, but I wouldn't think of all the things that might have popped up during the day.

Although the student quoted was perhaps not typical of the members

of the class, it seemed that the other students were quite involved with doing mathematics.

The student quoted in the previous paragraph seemed to be

especially stimulated by problems from the notes which were not

solved within a few days after they were brought up in class. He

seemed to be willing to work on them over a considerably longer

period of time than the other students would. With this longer

time to work, he was able to produce solutions to problems which no

other student had done. He did not present a large number of prob­

lems other than these. Thus, in a class with assignments to be

handed in after a certain time allowance, it is doubtful that the

instructor ever would have seen the evidence of this fr u itfu l work

even i f the student had pursued the problems to this length. In 97 a certain sense, the student seemed to be "marching to the beat of a d iffe re n t drummer." Fortunately, this was possible with the Moore method.

The preceding comments seem to provide some evidence for the following opinions of the writer and questions concerning the effects the teaching method had on the students' approaches to learning mathematics.

1.Even when a student did not get a solution, the work he did could be called "productive thinking." Did this contribute to his understanding of the mathematics being studied and his under­ standing of the solution when it was presented?

2. The students became quite involved with the mathematics and worked on the problems at odd moments. Is this more lik e ly to occur in a Moore method course than in courses taught by other methods?

3. The Moore method affords the students an opportunity to do creative work even i f i t takes them a long time to accomplish i t .

Concluding Remarks

A quote by Polya previously stated in Chapter I is worth repeating here: "... there are as many good methods as there are good teachers." [35: 114]. The question, then, seems to be:

How does one go about finding the unique teaching style best suited to oneself? The writer suspects that most teachers begin by imitat­

ing the style of one of their favorite professors. From this begin­ ning, by a trial and error process, they may eventually arrive at 98 something they can c all th e ir own. At least this has been the w rite r's experience. I t is perhaps also evident in the many former students of R. L. Moore who have adapted his teaching method to th e ir own courses.

It seems that it would be helpful to a teacher or prospec­ tive teacher to have available expositions of various teaching methods. Perhaps such expositons would provide the teacher an opportunity to experience vicariously teaching methods that he would not otherwise experience. It seems that this would provide him with a broader basis from which to develop his own style of teaching.

It is hoped that this dissertation has provided one such exposition.

Beyond exposition of teaching methods there is a need for studies which deal with various aspects of college teaching. Col­ lege teaching and preparation for college teaching have often been neglected areas. It often appears to be assumed that a graduate degree is necessary and s u ffic ie n t evidence that a person is going to be a successful college teacher. Anyone who has been a college student probably knows counterexamples to both implications in that statement. What then does determine whether a college teacher will be successful? Can he be given special training that will improve his chances of success?

In addition to questions raised earlier, some questions remain. Implicit in these questions are suggestions for further study. Foremost among these is the question of extrapolation.

How similar would the results be with different students, a differ­ ent course, a different college, or a different teacher? As this study was being completed, the writer was planning to use the Moore

method in two courses the next semester. One of these, Mathematics

kOk, Intermediate Real Analysis II, will have as members of the class

only students two, five, and seven from this study—a select group.

Expectations fo r the course are high in lig h t of the achievements by

these students in this study. The other is a course in complex analysis which w ill be populated prim arily by seniors who were student teaching

the previous semester. This should be interesting in light of their

background in education. Does the Moore method have d iffe re n t and

perhaps useful effects on students going into secondary school teaching?

Longer range plans by the author c all for use of the Moore

method with sophomore students. It is anticipated that the problem

of motivation w ill be more acute with students at the sophomore level.

Are there variations that w ill be necessitated in order to teach

students in lower division courses?

I t seems then that an exploratory study such as this one

does not re ally have an ending. Its total impact on the w riter w ill

perhaps be f e l t more in the future than i t has been f e l t to this date.

Re-evaluation of the writer's opinions will surely occur as he con­

tinues to visit with the students involved in this study and involves

other students in courses taught by the Duren version of the Moore

method. APPENDIX A

COURSE MATERIALS

100 101

Mathematics *»03 Intermediate Real Analysis I

Western Maryland College Fall Semester 1969-70

Instructor: Duren

INTRODUCTION

Since for many of you this w ill be your f ir s t mathematics course to deal almost e n tire ly with theory, and since the course w ill be taught in a manner somewhat d iffe re n t from that to which you are accustomed, I feel that I should explain how the course w ill be organ­ ized and what we are going to do. The primary objectives of the course are for you to learn about (i) mathematics—in particular, you will be expected to do some mathematics, and (ii) real analysis.

Organization of the Course. You will be handed dittoed notes from time to time which w ill lis t axioms (when needed), definitions and problems (these may take the form of questions, conjectures, propositions, directions to do a specific activity, etc.) about the subject matter under study. You are to consider each problem as a challenge to be solved by you. This may involve giving a proof, ju s tify in g your answer to a question, performing a numerical calcula­ tio n , finding an example or counterexample, formulating a d e fin itio n , e tc . You are on your honor not to consult text books, faculty members or other students for a solution. You may ask questions or discuss your solutions with me outside of class anytime, but I will not tell you how to work a problem before i t has been presented in class.

There w ill probably be some problems which you w ill not be able to solve, but if you can obtain even a partial solution to a special case, this w ill be b etter than skipping the problem. Also, i f you have worked on a problem, even without v is ib le success, you w ill be more likely to understand the solution when you see it pre­ sented. Other than common knowledge of elementary logic and set theory, solutions fo r the problems must rely only on the axioms, definitions or other statements from the notes. It is not necessary, however, to do the problems in the order that they are on the notes, i.e . you may use a result from a proposition which comes la te r than the one on which you are working i f you prove the later proposition first. Be careful to avoid circular reasoning. You will notice that there are very few examples or numerical problems. You should supply these for yourself whenever you need them. I f you need a theorem or d e fin itio n fo r a proof that is not in the notes, statewhat you need, fin is h the proof and, in the case of a needed theorem, try to prove i t .

The activities in class will consist of student presentations o f th e ir solutions to the problems. The problems w ill be considered in the order they are on the notes. Sometimes I will ask for a 102 volunteer and other times I will call on individual members of the class in an order which I will determine, i.e. you may be able to guess, but you will never know for sure, when your turn w ill come. Once a solution has been presented, any class member who has reserva­ tions about the solution will have a chance to express them. The student presenting the solution w ill have a chance to make further explanations or corrections if necessary. Members of the class who have alternative methods of proof, different solutions or conjectures to make may volunteer them. Feel free to ask questions of me or the class if necessary to clarify your understanding of any solved prob­ lem. Once a problem has been presented, you are encouraged to dis­ cuss it further outside of class and to do additional reading on the topic.

Your grade for the course w ill be based almost e n tire ly on your class presentations and discussion.

Other Activities. In addition to your classroom presenta­ tions, there may be a few hand-in assignments which w ill be graded and returned. Also occasionally, I may lead a class discussion or lecture on a topic not in the notes. There w ill be an hour examina­ tion on Friday, December 19, 1969, which w ill have emphasis on your knowledge of real analysis. This exam w ill count only a small fra c ­ tion of your grade. You will be scheduled individually for a mid­ term and fin a l interview with me in which we w ill discuss your pro­ gress and other aspects of the course. Your grade will be assigned prior to the last interview.

Best wishes fo r an enjoyable semester! 103

CHAPTER 1

REAL NUMBERS

§ 0 Undefined Terms

We assume the existence of a set J* called the real numbers or more simply numbers, two operations + and • called addition and multi piicat ion respectively; and a relation of equals, A = B means that A and B are names fo r the same thing.

§ 1 Properties of Addition

Axiom 1. For any numbers a and b there is a unique number a + b.

Axiom 2. For any numbers a and b, a+b=b+a.

Axiom 3. For any numbers a, b and c, (a + b) + c = a + (b + c ) .

Axiom h. There is a number 0 which has the property that 0 + a = a fo re a c h number a.

Axiom 5. For any number a there is a number -a which has the property that -a + a = 0.

D efin itio n 1.1 For any numbers a and b, a - b means a + ( - b ) .

1.1 Problem: Let a, b, c and x be numbers, i. I f x + a = 0, then x = -a . i i . -0 = 0 I i i. -(a + b) = -a - b iv. -(-a) = a v. I f a + b » a + c, then b = c. v i . The equation x + a = b has a unique solution for x which is x = b ■ a. v i i . I f x + a » a, then x = 0; v i i i . a - (b - c) = (a - b) + c

§ 2 Properties of Multiplication

Axiom 6. For any numbers a and b, there is a unique number a • b.

Axiom 7. For any numbers a and b, a • b = b • a.

Axiom 8. For any numbers a, b and c, (a • b) • c = a • (b • c ) .

Axiom 9. There is a number1 which is not equal to 0 and which has the"property that 1 • a = a for any number a. 104

Axiom 10. For any numbers a, b and c, a • (b + c) = (a • b) + (a • c ) .

NOTE: In the absence of parentheses we w ill always m ultiply before adding or subtracting. Axiom 10 can then be w ritten as a • (b+c) = a • b + a • c.

1.2 Question: Is it necessary to specify that I is not equal to 0? Using the 10 given axioms, is i t possible to show that 1 + 1 / 0?

1.3 Problem: Let a, b, c and x be numbers. i . a • 0 = 0 ii. a • (-b) = (-a) • b = -(a • b) i i i . (-1) • a ■« -a iv. (-a) • (-b) = a • b v. (-1) • (-1) = 1

D efinition 1.2 For any number a, a2 means a • a.

1.4 Problem: I f a and b are numbers, then i . (a + b) • (a - b) = a2 - b2 i i . (a + b )2 = a2 + 2(a • b) + b2 i i i . (a - b )2 = a2 - 2(a • b) + b2

1.5 Question: Is there a number x such that x • 0 - 1?

Axiom 11. For any nonzero number a there is anumber a"1 which has the property that a-1 • a = 1.

1.6 Problem: Let a, b, c and x be numbers. i. If x • a ■ 1, then x = a"1, ii. I"1 - 1 iii. If a • b = 0 , then a = 0 or b = 0. iv. If a • b 4 0, then (a • b )" 1 -a-1 • b"1. v. If a / 0, then (a-1) -1 = a. vi. If a • b ■ a • c, then b = c.

D efinition 1.3 For any numbers a and b with b 0, means a • b_1. We also write a/b or a? b for ®. b 1.7 Problem: Let a, b, c and x be numbers. i. I f b t 0, then x = a/b is the unique solution of the equation x • b ■ a. ii. I f a + 0, then a”1 ■ 1/a. iii. I f b • d 0, then (a/b) • (c/d) = (a • c)/(b • d ). iv. I f a • b + 0, then (a/b)"1 ■ b/a. v. I f b • c •d / 0, then a/b a d c/d " b ’ c * v i. If b • c ^ 0, then £ a • b c " c • b 105

v i i . I f b • d ^ 0 , then a_ £ _ a » d + b » c b + d b • d v i i i . If b • c / 0, then (a 7 b) -7 c = a ? (b * c ) .

§ 3 Properties of Order

Axiom 12.Some real numbers are called positive. For eachnumber a, exactly one of the following is true: 1) a = 0 2) a is positive 3) -a is positive

For any positive numbers and b, a + b and a • b are positive numbers.

1.8 Problem: i. If a is a number and a t 0, then -a ^ a. ii. 1 is a positive number.

Definition 1.4 2=1 + 1, 3 = 2+1, 4 = 3+1, 5 = 4+1, etc.

1.9 Problem: i. 2, 3, 4, 5, etc are a ll positive numbers. i i . 2 ^ 0 iii. 2 + 3 - 5, 2 • 2 = 4, 2 • 3 - 6, 4/6 = 2/3

Def i n i t i on 1.5 A number is a negative number i f and only i f i t is neither a positive number nor zero.

NOTE: We write "iff" for "if and only if."

1.10 Problem: i. -1 , -2 , - 3, etc. are negative numbers. ii. The product of two negative numbers is a positive number. iii. If a is a positive number, then a-1 is a positive number. iv. If a is a negative number, then a"1 is a negative number. v. If a is a positive number and b is a negative number, then a • b is a negative number.

D efinition 1.6 For any numbers a and b i. a>biffa-bis positive i i . aa.

1.11 Problem: For any numbers a and b, exactly one of the following statements is true: I) a = b 2) a > b 3) a < b.

1.12 Problem; Let a, b and c be numbers. i. I f a < b and b < c, then a < c. i i . I f a > b, then a + c > b + c. iii. If a > b, then c • a > c • b. 106

§ k Properties of Absolute Value

Def ini tion 1.7 For any number a, the absolute value ofa is given by T7i la if a > 0 or a = 0 |a| " \ -a if a < 0

NOTE: It is convenient to write a > b to mean a > b or a = b, a £ bto mean a < b or a = b, and a < b < c to mean a < b and b < c.

1.13 Problem: Let a, b, c and x be numbers. I a | >. 0, {a| = |-a|, Either a = |a| or a = - 1a|. - |a| >. a >_ | a |. i i i . Ix - bI

§ 5 Positive Integers

Definition 1.8 A subset S of numbers is defined to be a successor set iff (i) 1 is in S, and (ii) for any number x, if x is in S, then x +1 is in S.

1.14 Problem: Give several examples of successor sets.

Definition 1.9 N^ is defined to be the intersection of all successor sets of numbers. A number x is a positive integer iff x is in N. Thus, is the set of all positive integers.

1.15 Problem: £ is a successor set.

1.16 Problem: If x is a number and x < 1, then x is not in N_.

1.17 Problem: Let Sn be a statement fo r each n in N_. I f ( i) Sj is true, and ( i i ) fo r any k in £ , i f is true then S ^ j is true; then Sn is true for each n in N.

1.18 Problem: I f M is a nonempty set of positive integers, then M contains a least element.

1.19 Problem: I f m and n are positive integers, then m + n and m * n are positive integers.

6 Properties of Completeness

Definition 1.10 A number x is called an upper bound for a set A of numbers i f f there is no member of A which is greater than x.

Axiom 14. I f a non-empty set A of numbers has an upper bound, then there is an upper bound for A that is less than any other upper bound for A. This number is called the least upper bound for A 1.20 Define: lower bound, greatest lower bound

1.21 Problem: If a non-empty set A of numbers has a lower bound, then there is a lower bound for A that is greater than any other lower bound for A.

1.22 Problem: The set of a ll positive integers does not have an upper bound.

1.23 Problem: i f a and b are positive numbers, then there is a positive integer n such that n • a > b.

1.24 Problem: If A is a non-empty set of numbers with least upper bound b, b is not in A and £ is a positive number, then there is a number x in A such that b - e < x < b. < 1.25 Problem: There is a number x for which x2 = 2.

1.26 Question: What is a real number?

D efinition 1.11 For any numbers a and b with a < b, the segment (a,b) is the set of numbers to which x belongs i f f a < x < b and the interval [a,b] is the set of numbers to which x belongs i f f a <_ x < b.

D efinition 1.12 A collection G of subsets of R covers the number set A iff A is contained in the union of the members of G.

1.27 Problem: If a collection G of segments covers the interval [a,b] then there is a f in ite subset G1 of G which covers [a ,b ].

1.28 Problem: Suppose that fo r each n in ]£, l n is an in te rv a l, and that l n+! is contained in l n for each n in N. Then there is number xsuch that x is in l n for each n In N_.

1.29 Problem: In problem 1.28, the common part is an interval or else contains only one number.

1.30 Problem: In problem 1.28, "in te rv a l" can be replaced by "segment1

Def i n i t i on 1.13 A set M is inf ini te iff for any positive integer n, M has more than n elements.

Definition 1.14 A number p is a cluster point of a set M iff each seg­ ment containing p contains a member of M different from p.

1.31 Problem: If M is an in fin ite subset of [0 ,1 ], then M has a cluster point.

1.32 Problem: If P is a cluster point of a set M and S is a segment containing P, then the intersection of M with S is infinite. 108

§ 7 Simple Graphs

Definition 1.15 An ordered pair (a,b) of numbers is called a point. The first term a is called the abscissa of P = (a,b) and the second term b is called the ordinate of P. The number plane is the collec­ tion of a l1 points.

NOTE: The symbol (a,b) has been given two meanings. It is always clear from the context which meaning is intended.

D efin itio n 1.16 For any point (a,b) the v ertic a l lin e through (a,b) is the set of all points (a,y), and the horizontal line through (a,b) is the set of all points (x,b).

D efin itio n 1.17 For any point (a,b) and number m the line through (a,b) with slope m is the set of all points (x,y) such that y - b = m • (x - a).

Def ini tion 1.18 A graph is a non-empty subset of the number plane. A simple graph is a graph which intersects no v e rtic a l line more than once. Letters such as f, g, h, etc. are used to denote simple graphs.

Definition 1.19 For any simple graph g, the collection of all a abscissas of points of g is called the domain of g and the collec­ tion of all ordinates of points of g iscalled the range of g.

NOTE: If (x,y) is a point of a simple graph g, the ordinate y of (x,y) is also denoted by g (x ).

Definition 1.20 For any simple graph g and any x in the domain of g, g is said to have property (C) at x iff for each pair (Hi, H 2) of hor­ izontal lines with (x ,g (x )) between them, there corresponds a pair (V i, V2) of v ertic a l lines with (x ,g (x )) between them, such that i f P is a point of g between Vi and V2, then P is also between Hi and H2.

1.33 Complete the d e fin itio n : The simple graph g does not have property (C) at x means ....

1.34 Problem: If f * * {(l/n,l) | n * 1, 2, 3, •.•} U {(0,1)}, then f has property (C) at each x in f .

1.35. Problem: I f g = f U { (0 ,2 ) } , then g has property (C) at x = 0.

1.36 Problem: I f h * { (l/(2 n + 1), 1) | n * 0, 1, 2, ...)U {(l/(2n), 1/2) | n * 1, 2, 3, ...}, then h has property (C)at each x in h.

1.37 Problem: If k * h U { ( 0 ,2 ) } , then k has property (C) at x = 0. 109

1.38 Problem: I f f is the line through (a,b) with slope m, then f is a simple graph and for each number x, f has property (C) at x.

1.39 Problem: I f the simple graph h has property (C) at x, then there are vertical lines and V with (x,h(x)) between them such that the part of h between Vj ana V2 is bounded i.e . contained between two horizontal lines.

1.40 I f f is a simple graph and x is in the domain of f , thenthe following two statements areequivalent: i . f has property (C) at x i i . fo r each positive number e there corresponds a positive number 5 such that i f s is in the domain of f and |s - x| < 6, then |f(s ) - f( x ) | < e.

1.41 Problem: I f the simple graph g with domain the interval [a,b] has property (C) at each x in [a ,b ], then g is bounded.

1.42 If g has property (C) at x and g(x) > 0, then there exists a segment (a,b) with x in (a,b) and a positive number d such that g(s) > d if s is in (a,b) and the domain of f.

D efin itio n 1.21 For any simple graphs f and g with overlapping domains Df and Dg respectively:

i. f + g is the simple graph to which (x,y) belongs i f f x is in both D* and Dg and y = f(x ) + g (x ). i i . fg is the simple grapn to which (x,y) belongs i f f x is in both Df and Dg and y = f(x ) • g (x ). iii. if for some x in both Df and Dg, g(x) / 0, then f/g is the simple graph to which (x,yj belongs i f f x is in both Df and Dg, g(x) f 0 and y = f(x)/g(x).

Definition 1.22 Let the domains Df and Dg of the simple graphs f and g respectively be such that g(x) is in Df for some x in Dg. We define f[g ] to be the simple graph to which (x,y) belongs i f f x is in Dg, g(x) is in Df and y ** f(g(x)).

D efinition 1.23 For any simple graph f and number c, cf is thesimple graph to which (x,y) belongs i f f x is in the domain of fand y = c » f(x ).

1.43 Problem: I f f and g are simple graphs with overlapping domains Df and Dg , c is a number and x is a number in both Df and Dg at which both f and g have property(C ), then I. cf has property (C) at x ii. f + g has property (C) at x iii. fg has property (C) at x iv. if g(x) t 0, then f/g has property (C) at x. no

1.44 Problem: I f the simple graph g has property (C) at x, and the simple graph f has property (c) at g (x ), then the simple graph f [g] has property (C) at x.

1.45 Problem: If the simple graph f with domain the interval [a,b] has property (C) at each x in [a,b], then f attains a maximum value on [a,b] and f attains a minimum value on [a,b].

1.46 Problem: The interval [a,b] in 1.45 can be replaced with the segment (a ,b ).

1.47 Problem:"Has property (C)" can be omitted from 1.45.

1.48 Problem: If the simple graph f has property (C) at each x in the interval [a ,b ], f (a) < Oi and f(b ) > 0, then there is a number c between a and b such that f(c ) = 0.

1.49 Problem: I f the simple graph f has property (C) at each x in [a,b], f(a) 4 f(b ) and y is a number between f(a ) and f ( b ) , then there is a number c between a and b such that f(c ) = y.

1.50 Problem: I f the simple graph f with domain [a,b] has property (C) at each x in [a ,b ],the range of f is an in te rva l.

1.51 Problem: If the simple graph f has domain and range [0,1] and has property (C) at each x in [0 ,1 ], then there is a number a in [0,1] such that a = f(a).

1.52 Problem: If the simple graph f with domain R and property (C) at 0 s a tis fie s f(x + y) = f(x ) + f(y), then 7(x) = kx for some fixed number k.

§ 8 Slope

Definition 1.24 For any simple graph f with P a point of f and any number m, f has slope m at P i f f i. each two v e rtic a l lines with P between them, have between them a point of f d iffe re n t from P, and ii. if L is a line of slope greater than m containing P and i f K is a line of slope less than m containing P, then there exist two vertical lines Vx and V2 with P between them such that i f Q j* P is a point of f between \!x and V2, then Q is also between L and K.

1.53 Problem: If the simple graph f has slope m at P and m' f m, then f does not have slope m1 at P.

D efinition 1.25 For any simple graph f with slope m at P, the line through Pwith slope m is called the tangent line to f at P and m is called the slope of f at P. I l l

1.5^ Problem: If the simple graph f has property (C) at x, then there is a number m such that f has slope m at (x,f(x)).

1.55 Problem: If the simple graph f has slope m at ( x , f ( x ) ) , then f has property (C) at x.

1.56 Problem: If f is a simple graph with slope m > 0 at ( x ,y ) , then there exist two vertical lines and V with (x,y) between them such that if (s,t) / (x,y) is a point of f between Vx and V2, then f(s) < f(x) if s < x and f(x) < f(s) if x < s.

1.57 Problem: If the domain of the simple graph f includes the segment (c,d), f has slope at (x,f(x)) for each x in (c,d), c < e < d, and f(x ) f(e ) for each x in (c ,d ) , then f has slope 0 at e.

1.58 Problem: Suppose f is a simple graph, m is a number, and that (x,y) is a point of f such that each two vertical lines with (x,y) between them, have between them a point of f d iffe re n t from (x,y). Then the following two statements are equivalent. i. f has slope m at (x,y) ii. If e is a positive number, there exists a positive number 6 such that if s in in the domain of f and 0 < |s - x| <6 then | f(s ) - f(x ) - m| < e ,

Definition 1.26 Let the simple graph f have slope at some point. We define f 1 to be the collection of all points (x,y) such that f has slope at (x,f(x)) and y is the slope of f at (x,f(x)). The simple graph f' is called the derivative of f .

1.59 Problem: With appropriate restrictions on f and g (supply such) 1. (f + g)' - f' + g' 11 • (fg)1 = f 'g + fg* H i . ( c f ) ' = c f' for any number c Iv (f/g)' = (f'g - fg ')/g v. (f[g])' = f ' [g]g'

1.60 Problem: Suppose f is a simple graph such that 1. f has domain [a ,b ], 11. for each x in [a,bj, f has property (C) at x, iii. for each x between a and b, f hasslope at (x,f(x)), and iv. f(a) = f(b) =■ 0. Then fo r some c between a and b, f has a horizontal tangent at (c,f(c)).

1.61 Problem: If f and g satisfy i, ii, and iii of 1.60, f(a) - g(a), and f(b ) = g(b), then for some c between a and b, f'(c) ■ g'(c). 112

1.62 Problem: If f satisfies i, ii, and iii of 1.60 and f'(x) > 0 for each x between a and b, then for each Xj and x2 such that a < x ! < x2 < b, we have f ( x x) < f ( x g) .

9 Sequences

Definition 1.27 A simple graph with domain the set of positive integers is called a number sequence. The number sequence { (l» a i)» (2 ,a 2) , (3»a75 . . .} is also denoted by simply ax, a ,, a ,, . . . or by {aj}|=x *

NOTE: Simple graphs are also called functions (more precisely— real valued functions of a real variable).

Definition 1.28 Let f x, f,, f 3, . . . be functions. The set of ordered pairs {(1 ,fi)» (2,f2), (3»f3) > . . .} is called a function sequence. I t is also denoted by { f j J js j .

Definition 1.29 The number sequence {aj};=i has a sequential limit point p iff for any segment S containing p, there is a number N > 0 such that if n is any integer greater than N, then an is in S.

1.63 Question: Can a number sequence have more than one sequential limit point?

1.64 Problem: For any number sequence ( a j } j “ i> the following two statements are equivalent. * i* {a;}|s1 has a sequential limiting point i i . For any e > 0, there is a number N > 0 such that i f each of m and n is an integer greater than N, then |an - am| < e #

Definition 1.30 Suppose {fj}j= x is a function sequence such that the intersection of the Dj is not empty where Dj is the domain of fj and M is a subset of the intersection of the D|. Then {f|}| con­ verges polntwise to a function f on M means that f(x ) is the sequen­ tial limit point of the number sequence {fl(x)}j= x for each x in M.

1.65 Problem: I f { f j } j “ i is a function sequence each term of which is a function having property (C) on [a,b] and {fj}j2 x con­ verges pointwise to a function f on [a,b], then f has property (C) on [a ,b ].

D efinition 1.36 The function sequence { f j } |K1 converges uniformly to a function f on a subset M of the intersection of the D; means that theintersection of the D; is not empty and for any e > 0 there is a number N > 0 such that if n is an integer greater than N, then for each x in M, |f(x) - fn(x)| < e. 113

1.66 Problem: If { f j } | = i is a function sequence with each function having domain [a,b] and having property (C) on [a,b], which converges pointwise to a function f having property (C) on [a ,b ], then { f j } | ” i converges uniformly to f .

1.67 Problem: I f { f | } j“ x is a sequence of functions having property (C) which converge uniformly to a function f , then f has property (C).

Definition 1.37 Let h « {(1>a1), (2,a2), (3»a3), ...} and k = { ( 1 »n1) 1 (2 ,n2) , (3»n3) , . . . } be number sequences with the range of k a subset of N with nj < n2 < n, < . . . . The number sequence h[k] = {(1>ani]", (2,an2), (§,an3J, ...} is called a subsequence of h and is denoted by {an >k=. IS *

1.68 Problem: Suppose G is a set of functions, each element of which has property (C) on [a ,b ], such that for any e > 0 there s a 6 > 0 such that if each of x and y is in [a,b] with x - y| <6 and g is in G, then |g(x) - g(y) | < e. Then if \9 j} j= is a function sequence each term of which is in G, there is a subsequence o f{g j}j= j which converges to an element of G.

1.69 Problem: In the conclusion of 1.68, "pointwise" can be replaced by "uniform." 114

CHAPTER 2

METRIC SPACES

§ 0 Functions

In this chapter the term "function" will be used more generally than in Chapter 1. For any nonempty sets A and B, by a function f from A to B, denoted f:A -»■ B, we mean a set of ordered pairs such that for each x in some subset of A there is at most one y in B with the property that (x,y) is in f. The domain of f is the subset of A given by {x | x is the first element of a pair in f} and therange of fis the subset of B given by {y | y is the secondelement of a pair in f}. If the pair (x,y) is in f we also denote y by f(x).

§ 1 Spaces

Definition 2.1 A metric space (E,d) is a set E together with a func­ tion d:E x E -»■ j* which associates with each pair p,q of elements of E a number d(p,q) such that for any p,q ,r in E: i. d(p,q) > 0 i i. d(p,q) = 0 iff p = q iii. d(p,q) - d(q,p) iv. d(p,r) <_ d(p,q) + d(q,r) The elements p,q,r, . . . of E are called points of E and d is called the distance function or metric.

2.1 Problem: Give several examples of m etric spaces.

2.2 Problem: Any set can be made into a m etric spaceby defining d(x,y) = 1 if x 5s y and d(x,x) = 0.

Definition 2.2 For any positive integer n, the set of all n-tuples of numbers En = {(aj, a2, ... , an) | ai, a2 an in R} is called n-dimenslonal Euclidean space.

NOTE: E1 = JR, but the E used in definition 2.1 and in later statements without a superscript is an arbitrary set.

2.3 Problem: For any p = (ai , az, . . . , an) and q B (b i, b2> . . . , bn) in En, define d(p,q) « / ( a x - bx) ^ + (a2 - b2) 2 + . . . + (an - bn) 2. Then d is a metric for E.

2.4 Problem: For any numbers a^, a2 , . . . , an, b j , b2, . . . , bn we have Ia 2b! + a2b2 + . . . + anbn l 5. / a T ^ ^ a ^ + V T T + aj^ " / b j z + b2z + . . . + bn2 115

2.5 Problem: For any numbers a i, a2, ... , an, bt, bg, ... , bn we have /(ai + bx)* + (a2 + kg)2 + ... + (an + bn)* <_ /aPr~+~Tzr +T77~:rT^r + A>\ + b z* + . . . + bn*.

2.6 Problem: I f p ,q ,r are points of a metric space (E ,d ), then |d(p,r) - d(q , r) | £ d(p,q) .

Definition 2.3 Let (E,d) be a metric space, p be a point of E, and r be a positive number. The open sphere in E of center p and radius r is the subset of E given by {x | d(p,x) < r} and the closed sphere in E of center p and radius r is {x j d(p,x) r}.

Definition l . k A subset S of a metric space E is open iff for each point p in S, there exists an open sphere of center p which is contained in S.

2.7 Problem: For any metric space E: i. the empty set

2.8 Problem: In any metric space, an open sphere is an open set.

Definition 2.5 A subset S of a metric space is closed iff its complement CS in E is open.

2.9 Problem: Any fin it e subset of a metricspace is closed.

2.10 Problem:For any metric space E: I. the empty set is closed i i . the set E is closed i i i . the union of any collection of closed subsets of E is closed iv. the intersection of any collection of closed subsets of E is closed.

2.11 Problem: If A is a bounded closedsubset of j*, then A containg its least element. 116

Hour Test

I. (36 pts) Circle T if true, F if false. 1. For any numbers a and b, | a + b | >_ | a | - |b |. 2. If x is a positive number, then there exists a positive integen n such that nx is a positive integer. 3. The set of positive integers does not have a lower bound. A. If f is a continuous function on (0,1), then f attains a maximum on (0,1) 5. If x is a cluster point of a set M, then there exists a number sequence that has x as a sequential lim it point. 6. If a number sequence {an}n=i has a sequential limit point then there exists a positive number M such that |an| <_M for al1 an. 7. If a function sequence converges uniformly to a function f on a set M, then the function sequence converges point- wise to f on M. 8. If [a,b] is a subset of the domain of f and f'(x) 0 for a ll x in (a ,b ), then f(a ) < _f(b ). 9. If f has slope m at (x,y) and g has slope m at (x,y), then fg has slope mm at (x ,y ). 10. I f f '( x ) > 0, then there exists a segment S containing x such that f(s) > 0 for all s in S. 11. If f is a continuous function with range an in te rv a l, then the domain of f is an interval. 12. If f = {(n,1/n) | n in N}, then f has property (C) at 10.

II. (2A pts) Give a counterexample for each statement in (l) that you indicated was false. Write the number of the statement, then the counterexample.

I I I . (27 pts) Give an example of: 1. A successor set 2. A set of numbers that has a least upper bound, but does not contain its least upper bound. 3. An infinite collection of segments that cover [0,2]. A. A f in it e subcollection o f the segemnts given (3) that covers [0 ,2]. 5. A set that contains all of its cluster points. 6. A simple graph that does not have property (C) at 2. 7. A simple graph that has slope zero at each point. 8. A number sequence that does not have a sequential lim it point, but has a subsequence with 0 as a sequential limit point. 9. A m etric space (be e x p lic it about components) 117

IV. Defini tion. The function f has limit L at b iff given any positive number e, there exists a positive number 6, such that |f(x) - L| < e for all x in the domain of f satisfying 0 < |x - b| < 6. 1. (5 pts) Complete: The function f does not have lim it L at b i ff . . . 2. (8 pts) Write as complete and polished a proof as you can for the statement: If the function f has limit A at b and the function g has limit B at b, then the function f + g has 1imi t A + B at b.

V. Being honest and f a ir with yourself and with me, what grade do you feel that you have earned in this course? If you would like to explain or justify your reasoning, please do so.

VI. If you would not be satisfied to receive the grade that you listed in (V), change it to one that you would be satisfied with in light of the stated conditions. (I have been known to give students the exact grade they listed.) APPENDIX B

FORMS FOR DATA COLLECTION

118 INDIVIDUAL STUDENT RECORD

Student

Class Number Date

1. Presented problem number Weighted score

2. Presented alternate solution to problem number

3. Offered constructive comment on problem number

k . Made conjecture concerning ______

5. Present, but made no contribution to class.

6. Absent

7. Student seems to be: a.working and achieving

b. working, but not achieving c. unmotivated

8. Comments: CLASS RECORD

Class Number ______Date

1. Material Covered:

2. Hints Given:

3. Worth of Class Meeting:

4. General Mood of Students:

Frustrated Tense Apatheti c

Stimulated Exuberant

5. Additional Comments: 121

DIARY DIRECTIONS

I. Use this booklet to keep records showing for each date: 1. The amount of time spent working for this course (not including class time). 2. The numbers of the problems you are working on. 3. The numbers of the problems you solve. k . (Optional) Any short comments or reactions about the course or materials.

I I . At about three or four times during the semester, please w rite an essay which makes a substantial expression summarizing your feelings, reactions, philosophy, or whatever at that time. I have not set specific dates since I want you to w rite this when you are in a re fle c tiv e mood or have something to say, but please find the time. Reminders for this are scattered through the booklet and extra paper is provided at the end of the booklet for these essays. Attach additional sheets i f necessary.

I would especially like to have comments from you ( i) regarding your concept of the nature of mathematics; in p a rtic u la r, the role of numerical problem solving, axioms, definitions, proof and counterexample, the variety of proofs and altern ate approaches possible in developing mathematics and the creation of such, and (ii) regarding what and how you are learning from this course; in particular, its effects on your creativity and motivation. Please be completely honest and candid with your remarks. Your grade in this course or any other course w ill not be based on your diary e n tries; in fa c t, your grade for this course w ill be assigned before I read this diary.

Please return the completed diary to me in class December 19, 1969. Your complete cooperation in this w ill be appreciated. You may pick up your diary during the second semester if you want i t. Lowell Duren 122

Guidelines for Midterm Interviews

1. What are your vocational objectives?

2. Why are you taking this course?

3. Where does this course fit in with regard to your vocational objectives?

4. Where does this course fit in with regard to your total under­ graduate program?

5. Do you lik e the course?

6. What are some specific things that you like?

7. What are some specific things that you dislike?

8. How can I improve the course?

9. How can I improve the course for the rest of the semester?

10. How can I improve the course fo r next year?

11. Do you lik e the teaching method?

12. Is the approach d iffe re n t from what you had previously experienced?

13. I f you had known that the course was going to be taught in this manner, would you have enrolled in it? l*t. What have you learned?

15. Do you feel that you are learning more or less than in a more conventional lecture-discussion course? with respect to real analysis? with respect to mathematics? with respect to teaching- learning?

16. How do you rate (grade) yourself fo r the course to this date?

17. Do you have anything that you would like to "get o ff your mind" at this time? 123

Guidelines for Final Interviews

1. What is your overall reaction to the course?

2. Do you have comments on some specific instances from the course?

3- What part of the subject matter did you like the most? least?

4. To what extent did you attempt to recall the proofs of fa m ilia r statements from previous courses.

5. Did this course a ffe c t your view of "what is mathematics?"

6. Do you see mathematics as an activity?

7. What is the role of numerical problem solving, axioms, defini­ tions, theorems, proof, and counterexample in mathematics?

8. Did this course affect your ability to write proofs?

9. Did this course a ffe c t your understanding of what a proof is?

10. Did this course a ffe c t your a b ility to read mathematics?

11. Did this course a ffe c t your creativity?

12. Did this course affect your ability to create your own proofs?

13- Did this course a ffe c t your confidence in your creative a b ility ?

14. Did this course a ffe c t your attitude toward learning mathematics?

15* Do you experience a sense of accomplishment when you complete a proof?

16. Do you feel motivated to learn more real analysis on your own?

17- Do you think that I should use this teaching method in my other courses? which ones?

18. What modifications in the teaching method would you recommend?

19- Should students be allowed to work together? use texts?

20. To what extent is the success or fa ilu re of this method of teaching dependent of the instructor?

21. Do you consider the course to have been a success or fa ilu re for yourself? for the class? 124

22. Did you spend more or less time on this course than you usually do on an advanced mathematics course?

23- What was your primary motivation for studying?

24. To what extent were you influenced by the other members of the class?

25. Do you feel that your grade accurately reflected your work and achievement in the course?

26. Do you have any questions that you would like to ask me or additional comments that you would like to make? APPENDIX C

COURSES AND CATALOG DESCRIPTIONS

125 126

109 Introduction to College Mathematics

A unified treatment of the basic properties of algebra and trig­ onometry with p a rticu la r emphasis upon the nature of mathematics as a logical system; in it ia l study of sets, the real number system, and the properties of the field of real numbers; brief review of elementary algebra; intensive study of circular, linear, quadratic, polynomial, exponential, and logarithmic functions.

113 Analytic Geometry

A study of the line, circle, conic sections, curves and curve sketching, polar coordinates, and parametric equations.

114 Calculus I

The fundamental formulae of differentiation and integration with thei r applications.

115, 116 Calculus II, III

Definite integrals and applications, series, expansion of functions, hyperbolic functions, partial differentiation and applications, multiple integrals.

204 D iffe re n tia l Equations

A study of equations of order one and degree one, with applications; equations of order one and higher degree; linear equations with constant coefficients; the Laplace transform.

221 Fundamental Concepts of Algebra

An introduction to modern algebraic theory; emphasis on the nature of the structures of algebra, including groups, rings, fields and vector spaces; selected topics from elementary number theory, polynomial theory and matrix theory.

222 Fundamental Concepts of Geometry

Foundations and evolution of geometry; selected topics from Euclidean and non-Euclidean geometries, projective geometry, affine geometry; studies in the nature of proof and famous geometric problems. 127

307 Abstract Algebra

An introduction to modern algebraic theory, rings, fields, poly nomials over a fie ld , algebra of matrices. [Not offered a fte r 1968-69.]

308 History of Mathematics

A study of mathematics from p rim itive counting systems to the development of modern mathematics, with p articu lar emphasis on the seventeenth century.

309 Linear Algebra

The theory of finite-dimensional vector spaces, linear transfor­ mations and matrices, with geometric applications.

311 Topology

Introduction to set theory; topological spaces, product spaces; lim it points, open and closed sets; countability axioms; separa­ b ilit y ; continuous mappings and homeomorphisms; v arieties of compactness; separation axioms; v arieties of connectedness; metric spaces.

317 Abstract Algebra

A rigorous presentation of the theory of groups, rings, and fields through a study of selected topics, with emphasis on the study of groups; homomorphisms and isomorphisms of groups and rings; iso­ morphism theorems; Sylow theorems; ideals; Galois theory.

323 Probability

A study of sample spaces, combinatorial analysis, conditional p ro b a b ility , Bayes' Theorem, random variab les, Chebyshev's Theorem, binomial distributions, and applications.

32A Probability and Statistics

A study of- probabi1ity spaces, random variables, confidence inter vals, central lim it theorem. [Not offered a fte r 1967-68.] 128

324 Mathematical Statistics

A study of measures of central tendency, statistical estimation, confidence intervals, linear correlation, applications of prob­ ability theory, and other selected topics.

325 Projective Geometry

A study of synthetic projective geometry, including the projective plane, incidence relatio n s, harmonic sequences, projective trans­ formations, and the principle of duality; selected topics from analytic projective geometry, including transformations, cross ratios, and conics; the theorems of Desargues, Pappus and Pascal.

352 Research Seminar I

A review of research techniques specifically applied to a project in mathematics which w ill be developed into a w ritten seminar paper. This course is open only to juniors who expect to continue their research into the senior year.

403 Intermediate Real Analysis I

A rigorous study of infinite sets, functions, limits, continuity, derivatives, and Riemann integrals.

404 Intermediate Real Analysis II

A continuation of Mathematics 403; a rigorous presentation of sequences and series of real numbers; topics selected from metric spaces, elementary functions, sequences and series of functions.

416 Complex Analysis

An introductory course in the theory of the functions of a complex variable.

451 Integration of College Mathematics

A seminar in which the scope of collegiate mathematics is explored through problems and discussion of selected topics. APPENDIX D

EXAMPLES OF DATA CONCERNING PRIOR COURSES

129 130

EXAMPLES OF DATA CONCERNING PRIOR COURSES

Example One

Course: 113 Analytic Geometry

Instructor: McDonnell

Year and Semester: 1966, summer

Text: Middlemiss, Analytic Geometry

Students: 6, 8

Teaching Method: Not available

Instructor's Outline: Not available

Final Exam: Not available

Example Two

Course: 114 Calculus I

Instructor: Amoruso

Year and Semester: 1966-67, II

Text: Taylor and Wade, University Calculus

Students: 2, 4, 6, 8

Teaching Method: Not available

Instructor's Outline: Not available

Final Exam:

1. Evaluate each of the following: (a)

2. Draw a figure and find the total area bounded by the following curves: y = x2 - 2x - 3, y = 0, x “ -3, x = 1 3. Determine the interval over which F(x) is increasing and the interval over which F(x) is decreasing: F(x) ■ 3x5 - 5x3. 4. Find the equation of the normal to the curve: x2 + xy + y2 =» 3 at (1,3) 131

5. Given that the height of a stone above the ground is given by: s = 80t - 16 t2 - 96. Find (a) its in it ia l v elo c ity , (b) the time it will take to reach the greatest height. 6. (a) Given that G(h) = 1/x - 1/(x + h) , find the lim it of h G(h) as h + 0. (b) Solve the following equation: |x - 4| = 11. 7. Given that F(x) = (x - \)Jlx + 4x, find F1 (x ). 8. A carpenter wishes to build a box with a square base, but open at the top. Find the dimensions of the largest box he can build, i f he has available 48 square feet of lumber. 9. Determine a, b, c, and d so that the curve y = ax3 + bx2 + cx + d has a c r itic a l point at (1 ,-2 ) and touches the lin e 3x + y = 0 at (0 ,0 ). 10. The firs t quadrant area bounded by the coordinate axis and the parabola y2 + 4x = 16 is revolved about the x-axis. Find the volume of the solid generated.

Example Three

Course: 116 Calculus I I I

Instructor: Spicer

Year and Semester: 1967“68, II

Text: Taylor and Wade, University Calculus

Students: 4, 6, 8

Teaching Method:

Lecture-questions. Instructor presents a number of examples of problems and solutions. He feels that the students will learn the necessary theory through examples and problem solving; hence, the emphasis is on problem solving rather than theory.

Instructor's Outline: Not available

Final Exam:

1. Replace the iterated integral by an equivalent one where the order f 4 / /4 -x of integration is reversed: I l , f(x,y) dy dx Jo) 2 (]/2 x - 1 2. Obtain the value to three decimal places for I L dx JO x 132

3. Find the complete interval of convergence fo r: (x-2) - 2(x-2)2 + 3(x~2)3 **. If w= (x2 + y2 + z2)-1, find 32w/3z3x. 5. Expand F(x) = 1/x in powers of x-3 to 3 terms. 6. Evaluate limit (sec x - tan x) x-nr/2 7. Find the volume of the region in 3~space bounded above by z = x2 + y2 , below by z = 0, and on the sides by x2+ y2 = 1. 8. Find the mass of a plate bounded by one arch of the curve y = sin x, and the x-axis, if the density is proportional to the distance from the x-axis. 9. Given u = (x2 + y2)/xy, x = ercos s, y = ersin s, find 3u/3s. 10. Find the integral of the function f(x ,y ) = x cos (x+y) over the triangle whose vertices are (0,0), (tt,0) , and (ir,ir).

Example Four

Course: 309 Linear Algebra

Instructor: Sorkin

Year and Semester: 1968-69, II

Text: Paige and S w ift, Elements of Linear Algebra

Students: 1, 2, k, 5, 6, 8

Teaching Method:

Lectures from prepared notes which do not necessarily follow the order of the text. Students are asked questions in class during lecture and work on problem sets fo r homework.

Instructor's Outline:

1. Vectors in the plane II. Vector space III. Subspace IV. Span, lin e a rly dependent, lin e a rly independent V. Dimension, basis VI. S - T, sn T V I1. Linear transformations V I I I . Range, null space IX. Compos i te X. Invertible operator XI . Similar matrices X I1. Determinants X I11. Proper vector, proper values 133

XIV. Characteristic polynomial of matrix A, operator T on Vn(R) XV. Diagonalizable operator XVI. Inner product, orthogonal, norm of a vector XVII. Cauchy-Swartz inequality

Final Exam:

1. (a) Define: V is a real vector space. (b) Define: X j, X2 , . . . , Xn are lin early independent vectors. (c) Are the vectors Xx = [1,2,3], X2 = [0,0,0], X3 = [^,5,6] lin ea rly independent? Explain. (d) Define: The vectors Xlf X2 ...... Xn span V. 2. (a) Define: D is an inner product for the vector space V. (b) Show that D(X,Y) = x 1y1+ (x x + x2) ( y 1 + y2) fo r X = [x^Xg] and Y « [ylty2] is an inner product on V2(R). (c) Using this inner product, find |z| for z = [5,3]* 3. Suppose V and W are real vector spaces and T:V -*■ W is a linear transformation. (a) Prove that the range of T is a subspace of W. (b) Let {Xlf X2...... Xn) be a basis for V. Prove that the set (TXj, TX2...... TXnJ spans the range of T. A. Suppose T[x,y,z] = [x+y-2z, x-3y+2z, 5x-3y~2z]. Find a basis fo r the null space of operator T. What must be the dimension of the range of T? 5. Suppose T[1,1] = [2,-1] and T[2,3] = [-1,3]. Find [T]{e<},

the matrix of T with respect to the natural basis. 6. Prove one of these theorems: (a) If S = {Xx, Xx, ... , Xn) is an orthogonal set of non-zero vectors in inner product space V, then S is a lin e a rly independent set. (b) Let T be a linear operator on Vn(R).Suppose Xa, X2, ... , Xm are distinct real proper values for T. Let Xj be a proper vector associated with Xj for each i = 1, 2, ... , m. Then Xj, X2, ... , Xm are linearly independent. 7. Let M be the set of a ll 2 x 3 matrices [a b cl where a ,b ,c ,d ,e ,f Id e f] are real. (a) Define addition and scalar multiplication to make M a real vector space. (b) What is a basis for M? (c) What is the dimension of space M? (d) Make up an isomorphism from M to Vn(R) fo r some n. 8. Suppose T[x,y] = [5x-y, x+3y]. Is T invertible? If no, why not? If yes, find T"1[x,y]. 134

^ I U/m.i /» t I/M ■ 4-/»1 1 4- U n4 < u _ O n _1_ 0 > * «• A U <%r> m/\ t\/s»%*■ P j y | ^ |

nontrivial solution? 10. (a) Define: T is a linear transformation from V to W. (b) Let W = the vector space of a ll polynomials ofdegree less than or equal to 3. Show that T[a1,a2,a3] = ajX + a2x2 + a is a linear transformation from V3(R) to W. (c) Is T onto V? Explain. (d) What is a basis for the range of T? (e) Is T one-to-one? Explain. 11. Suppose S = L{ [3,6,1], [2,1,1]} and T = L{[-1,0,1], [2,3,1]} are 2 subspaces of V3( R). Find a basis fo r % (\ T. What must be the dimension of S + T? Extra c red it: Prove the other theorem in #6, or state and prove the Cauchy-Schwartz Inequality. APPENDIX E

BACKGROUND OF STUDENTS

135 136

Student One

Year Sem. Course Instructor Grade

1966-67 1 115 Calculus II Spicer A

1966-67 11 116 Calculus 111 Spicer B

1967-68 1 307 Abstract Algebra Lightner B

1967-68 11 308 History of Mathematics Lightner B

1967-68 11 324 Probabi1i ty and Spi cer B Stati sti cs

1968-69 11 309 Linear Algebra Sorki n C

1968-69 11 352 Research Seminar Lightner B

1969-70 1 403 Intermediate Real Duren C Analysis 1

1969-70 1 451 Integration of College S ta ff B Mathemati cs

Comments from instructors on writing proof:

Fair to better than average.

Better than average. Confused at times--knows when going wrong, but not how to get out of it.

Quite poor in beginning, but coming around. Has potential for being a good student. 137

Student Two

Year Sem. Course Instructor Grac

1966-67 1 113 Analytic Geometry Amoruso B

1966-67 II 114 Calculus 1 Amoruso C

1967-68 1 115 Calculus 11 Spi cer C

1968 sum. 222 Fundamental Concepts Lightner A of Geometry

1968-69 1 116 Calculus 111 Spicer B

1968-69 1 307 Abstract Algebra Duren B

1968-69 11 204 Differential Equations Jordy B

1968-69 11 309 Linear Algebra Sorkin B

1968-69 11 352 Research Seminar Lightner A

1969-70 1 323 Probabi1? ty Jordy C

1969-70 1 325 Projective Geometry Jordy A

1969-70 1 403 Intermediate Real Du ren B Analysis 1

1969-70 1 451 Integration of College S ta ff B Mathemati cs

Comments from instructors on writing proof:

Fair to good— talks a better game than he w rites.

Good at speaking up during proofs in class. Spots holes and is able to plug them.

Good, asks good questions and suggests ways to go. Better at this than putting on paper.

Good, can understand proof and figure out next step. 138 Student Three

Year Sem. Course Instructor Grade

1967-68 1 113 Analytic Geometry Amoruso A

1967-68 1! 114 Calculus 1 Amoruso A

1968-69 1 115 Calculus II Sorki n A

1968-69 1 221 Fundamental Concepts Lightner A of Algebra

1968-69 II 116 Calculus III Spicer A

1969-70 1 317 Abstract Algebra Duren A

1969-70 1 403 Intermediate Real Duren A Analysis 1

Comments from instructors on writing proof: o Good, one of the best in class.

0 139 Student Four

Year Sem. Course Instructor Grade

1966-67 1 113 Analytic Geometry Amoruso B

1966-67 II 114 Calculus 1 Amoruso B

1967-68 1 115 Calculus 11 Spi cer B

1967-68 II 116 Calculus ill Spicer B

1967-68 11 222 Fundamental Concepts Lightner A of Geometry

1968-69 1 307 Abstract Algebra Duren B

1968-69 1 323 Probabi1ity Jordy A

1968-69 II 204 D iffe re n tia l Equations Jordy B

1968-69 II 309 Linear Algebra Sorkin B

1968-69 11 324 Mathematical Statistics Jordy A

1968-69 11 352 Research Seminar Lightner A

1969-70 1 317 Abstract Algebra Duren A

1969-70 1 403 Intermediate Real Duren C Analysis 1

Comments from instructors on writing proof:

Sometimes has trouble with proofs. Makes an attempt but doesn't seem to understand quite what to do.

Good on proofs.

Only one in class that consistently knew what was going on.

Good, but disappointing. Knows what a proof is about. Student Five

Year Sem. Course Instructor Grac

1966-67 1 1U Calculus 1 Denni s A

1966-67 II 115 Calculus II Denni s A

1967-68 1 116 Calculus III Spi cer A

1967-68 II 222 Fundamental Concepts Lightner A of Geometry

1968-69 1 307 Abstract Algebra Du ren A

1968-69 1 311 Topology Sorki n B

1968-69 II 204 Differential Equations Jordy A

1968-69 11 309 Linear Algebra Sorki n A

1968-69 II 352 Research Seminar Lightner A

1968-69 II 416 Complex Analysis Duren A

1969-70 1 317 Abstract Algebra Du ren A

1969-70 1 323 Probabi1i ty Jordy A

1969-70 1 403 Intermediate Real Duren A Analysis 1

1969-70 1 451 Integration of College Staff B Mathematics

Comments from instructors on writing proof:

Very good. Knows when he has a proof and when he d o e s n 't.

Good.

Some trouble. Sometimes thinks he is proving something that he isn't. Student Six

Year Sem. Course Instructor Grac

1966 sum. 109 Introduction to College Spi cer B Mathematics

1966 sum. 113 Analytic Geometry McDonnel1 B

1966-67 11 114 Calculus 1 Amoruso C

1967-68 1 115 Calculus II Spi cer D

1967-68 1 1 116 Calculus 111 Spi cer C

1967-68 11 222 Fundamental Concepts Lightner 0- of Geometry

1968-69 1 307 Abstract Algebra Du ren D CO CM 1968-69 1 Probabi1i ty Jordy D

1968-69 11 308 History of Mathematics Lightner C

1968-69 1 i 309 Linear Algebra Sorki n D

1968-69 11 352 Research Seminar Lightner B

1969 sum. 324 Mathematical Statistics Jordy C

1969-70 1 325 Projective Geometry Jordy C

1969-70 1 403 Intermediate Real Duren C Analysis 1

1969-70 1 451 Integration of College S ta ff B Mathematics

Comments from instructors on writing proof:

Attempts do not make sense.

Very weak in doing proofs.

Nothing.

In a fog. Tries, but gets upset easily. Has trouble with definitions, let alone proofs. 142

Student Seven

Year Sem. Cou rse Instructor Grad

1967-68 11 114 Calculus 1 Amoruso A

1968-69 1 115 Calculus II Sorki n A

1968-69 1 221 Fundamental Concepts Lightner A of Algebra

1968-69 11 116 Calculus III Spi cer A

1968-69 11 222 Fundamental Concepts Lightner A of Geometry

1968-69 II 323 Probabi1i ty Jordy A

1968-69 11 352 Research Seminar Lightner A

1969 sum. 324 Mathematical Statistics Jordy A

1969-70 1 311 Topology Sorki n A

1969-70 1 325 Projective Geometry Jordy A

1969-70 1 403 Intermediate Real Duren A Analysis 1

1969-70 1 451 Integration of College S taff B Mathemati cs

Comments from instructors on w ritin g proof:

No problem with w riting proof. Did a ll the extra credit s tu ff.

Good. One of the best in class.

Very good on originals. Tries until he gets somewhere.

Understands d efin itio n s and proofs. Knows when pieces are put together. Student Eight

Year Sem, Course Instructor Grade

1966 sum. 109 Introduction to College Spicer C Mathematics

1966 sum. 113 Analytic Geometry McDonnel1 D

1966-67 11 114 Calculus 1 Amoruso D

1967-68 1 115 Calculus II Spi cer D

1967-68 11 116 Calculus III Spi cer C

1968-69 1 307 Abstract Algebra Duren D

1968-69 1 323 ProbabiIi ty Jordy D

1968-69 11 222 Fundamental Concepts Lightner C of Geometry

1968-69 11 309 Linear Algebra Sorki n D

1968-69 11 352 Research Seminar Lightner B

1969-70 1 325 Projective Geometry Jordy B

1969-70 1 403 Intermediate Real Duren C Analysis 1

1969-70 1 451 Integration of College S ta ff B Mathematics

Comments from instructors on w riting proof:

Attempts proof, but doesn't seem to understand what i t is a ll about.

Very weak in doing proof.

Can't do anything.

Seems to have no understanding. APPENDIX F

STUDENT ESSAYS STUDENT ESSAYS

The following essays were w ritten by the students at various times during the course as a part of the record keeping in their d iaries.

Student One

September 28

To this point in the course I feel satisfied with my achieve­ ment and success. I find that I can solve most of the problems. How­ ever, when I cannot solve a problem I find that I feel frustrated and stupid.

I enjoy the classroom atmosphere. At f i r s t , I was embarrassed in class when I didn't have the solution to a problem. But after I realized that I wasn't the only one who couldn't prove all of the prob' lems, I'v e become more at ease.

The tape recorder bothered me on the first few days but I've found that I completely forget about i t when I'm in class.

I enjoy the class more than any other math course I've taken at WMC because I don't feelso pressured. I never feel as though I'm put on the spot to give an immediate answer.

I don't find myself bored in this class as I often become in some lecture courses.

I've often felt a little awkward or inadequate when standing in front of my peers in a math class. But in this course, I feel more at ease. I think the experience will be good for my teaching career.

W ell— so fa r, so good!

October 19

I'm afraid that I'm beginning to feel alittle frustrated in this course due to the fact that I find I can't prove many of the prob lems. Sometimes I think that I'm either intellectually inferior to my classmates or I have a mental block for analyzing proofs. I think the fact that I'n not a whole-hearted math major (half my heart is in the English department), has a lot to do with it. Probably as a result of my English courses, I haven't developed my mind to function as ana­ lytically as most math majors. As a matter of fact, I generally feel 146 that my abilities in the field of literature exceed my abilities in mathemati cs.

Anyhow—back to real analysis (whatever that is)--1 often feel that my attempts at problem solving are in vain. Sometimes I can imagine you s ittin g there thinking how unconcerned I am about the course when I re a lly have made an honest attempt.

I'm constantly trying to analyze my mind to figure out why I carft see through through a proof. I think that i t has something to do with my f i r s t college math courses with Dr. Spicer who put abso­ lutely no emphasis on theory. He trained his calculus students to follow formulas without an actual understanding of the theory behind them.

At any rate, I'm still trying.

November 23

I thought I was frustrated when I wrote my last essay, but I'm even more frustrated now. At this rate, I'm going to develop an in fe r io r ity complex!

I find that I can usually understand the proofs when I see them presented in class. But when I attempt to solve the problems, I often have no idea where to begin. I've also had some trouble understanding definitions lately which is undoubtedly one of the reasons why I can't see through a proof c le a rly.

Do you think I'm a hopeless mathematician?

December 16

Well, I've ascended from my pit of frustration in the last essay since I'v e found that I can understand and solve problems on metric spaces. It's amazing what a little success can do for one's ego! However, I have a feeling that my ego w ill be irreparably deflated when you hand us back the test on which I'm certain I did rather poorly.

I really managed to get myself in a state of anxiety over that test! I don't even want to speculate on what my nervous state w ill be before I take comprehensives!

Anyhow, I feel that I'v e learned a great deal through this course not only about real analysis but also about myself and my a ttitu d e towards mathematics. I re ally think that I could have learned more real analysis if I had had time to get individual help from you throughout the semester. However, with a double major and working in the dining hall, I run on a pretty tight time schedule 147 throughout the week. So I rarely had time to stop in for help and guidance.

I'd like to commend your teaching method in this course— I feel that the course was extremely successful. I can honestly say that I enjoyed the course although my achievement was not what I had expected in the beginning or what I would have liked i t to be.

As a professor, I rate you as one of the best at Wimcee. I think that your concern for individuals is unequalled by most teachers. (I'm not saying this to be schmultzy or to be a brown-noser!) I re ally enjoyed the course. I hope that you have as much success in your future years of teaching. Good luck on your dissertation!

Merry Christmas!

Student Two

October 16

This course in terrelates not only analysis but also group theory and quite a bit of simple arithmetic. By'looking at some­ thing you did and accepted and now you are trying to prove that which you accepted as true such as 2 + 3 = 5» 3 + 4 = 7» etc. Mathe­ matics is the building of these basic axioms we have been given.

This course is good in that not only do you approach the things you accepted as true and prove them. This is also interesting in that there are sometimes more than one way to work these problems. Because of the math integration course, I tend to look at mathematics more in views of philosophies. I feel that I am more of an intui- tio n is t than a fo rm alist. Some of these problems I can see in tu i­ tively but find it hard to prove the point and show a rigorous proof of such.

December 2

Now that the course has progressed even more, I am convinced that many phases of math are being taken into consideration, such as the topology proof that [student five] did. There is also some anal­ ytics with the work with slopes and continuity. Even more I can see intuitively many of these problems, but still can't get a rigorous proof of some of them.

The course has bogged down some, but this is because of the new approach to continuity and derivatives which is really slopes. 148 Student Three

October 9

You could say I'm frustrated at this time. It seems that every time I sit down to work analysis problems, I get nowhere. You feel you have wasted time when you scratch for an hour and accomplish nothing.

I do find it helpful when the professor leads the class. For example on 10/8 when we fin a lly proved #1.18. I think you can only look at a thing from so many angles; a fte r that you waste your time. A fter everyone works on a given problem for a week or two, I think that to save wasting any more time the professor can lead the class. From a strictly personal point of view, either I figure a problem out or I can't; if I can't it doesn't make any difference if another student shows me how to do i t or the teacher shows me.

I would have to say that I like the format of 403•Generally I find the class more interesting than say Math 317. By having all the problems ahead of time, you can work at your own speed. This is especially convenient when you get snowed under by tests in other subjects.

November 21

It seems that the more I see of the course, the better I like it. The class is relaxed and informal, yet at the same time I feel I'm learning a lot. I find that I get much more out of a proof I work on 40 or 70 minutes than one that is thrown at me cold in class, as is usually the case in a course like abstract algebra. Even if you work for an hour and don't prove anything, you get to see the complications and better appreciate the proof when you finally see it presented. This hour of "fruitless" work might also enable one to see a loophole in another's proof.

I have a tendency to not work regularly on analysis. (This is readily evident in the daily record.) I don't think this is nec­ essarily bad, as long as I keep up with the work. There is the prob­ lem that when you work too fa r ahead you forget the material and have difficulty presenting it.

I still find that if I work for any length of time and accomp lish nothing there is a tendency to put the book away and forget analysis. I usually quit because there are 2 or 3 or 4 or . . . problems fo r which I can get nothing.

December 3

I find that there is a definite lack of numerical examples in 149 our outline. For example, if there were a couple of examples of functions that converge pointwise (Def'n 1.30) i t would give the student a framework to work w ithin. The last place we had concrete illustrations was when discussing "prop (C)“ (#1.34 “ 1.37)• When a student tries to come up with examples out of his head, he might not think of all the various possibilities, e.g. he might thinkof (1/2)1 approaching 0 as i but he might not think of (-1/2)' approaching 0 with some values positive and some negative values. I frequently tend to lose generality in coming up with examples.

December 18

The inconsistency in the amount of time I spent on this course is shown in Figure 1 (enclosure).

Generally, I'm glad I took this course. I've never been involved in something like this and have enjoyed i t . I also think I've learned a great deal.

Student Four

September 30

The f i r s t 13 problems provide us with a chance to actually see that all these properties of numbers could be proved, not just believed to be true. The format of the class seems to lend it s e lf well to the course so fa r.

This early material provides a stronger foundation in showing us exactly why all of these properties known for so long are true.

November 12

I am not having success lately in proving, problems and feel somewhat discouraged. My creativity is not very great and I can't seem to develop it much either. At the moment the structure does not seem to help my learning too much.

I now see it is much more difficult to formulate a proof than it appears in any textbook. Creativity in developing proofs certainly appears to be at the heart of this mathematics course. Many of these proofs look nice and somewhat easy once you see how to work i t . This course d e fin ite ly shows there is more to being a mathematician than ju st being able to apply math. The basic types of proofs being used I understand; the problem is picking the rig h t one. 150

Student Five

September 26

At this point in the course I must say that your treatment of the material has been very good. This treatment has its strength in the classroom discussions which are so noticeably lacking in many courses. Too often the professor gives out information and the students merely absorb it. The discussion among students and professor generates many more insights into a particular problem and these added insights multiply into other insights for other problems. In addition to this, the discussion makes the subject more interesting and alive to the student. I'm not sure if it does this for everyone, but I know that it does it for me. In essence, the point that "the more closely a student is involved in his sub­ ject, the more he learns," seems to hold in this case.

From another angle, this dialogue concerning a problem has shown me the workings of another person's mind in what to do and how to accomplish the desired result than ever before. Too often a prof­ essor will write a proof on the board completely out of his head. This method doesn't explain to the student why he decided to approach the proof using one axiom as opposed to another. In our seminar type of system we can see more clearly why a person chooses a p articu lar course of action. He may explain that he trie d one method and found i t fru itle s s and then went on to a b etter method. In the classroom, the student gets only the final method which may be less important in the long run than the attempts that were fru itle s s .

Many times I have found i t d if f ic u lt to remove the notions that I have about real numbers. For instance when we attempted to prove that 1 is a positive number my ability to think clearly in relatio n to the given premises was clouded by a ll of the notions I had about 1 that had not been developed previously in the course.

Thus, it is valuable experience to begin a mathematical system from "scratch" as we have done with the real numbers. This rigorous approach enables us to gain experience in working with a system in which the only things we can assume are the given state­ ments or the ones we have proved previously. This experience can be then applied to other types of mathematical structures. In any mathematical system assumptions distasteful . F in a lly , this course helps the student to remove assumptions and work from a more firm logical basis.

October 29

At this point I feel that some good things have been happening but there some things that aren't so good. 151 We are certain ly learning various approaches to problems as well as general ideas in writing any proof. In addition we're learning that a p articu lar type of approach may be applicable in several instances along the way.

There are aspects of this method of running the course that bother me. It seems as if we stagnate quite a bit in class. For example in one recent class we worked on one problem and only got part of a solution. I realize that the purpose of the course is not merely to crank out results. I f this were a ll we wanted we could do just as well by going to a lecture format.

The things such as approaches to a proof, generalized concepts and depth in working a problem are extremely valuable. Thus, we shouldn't scrap the seminar in favor of the lecture. At any rate, I think we should be covering more analysis than we have been covering. What I'm getting at is that I question whether the results we obtain actual ly warrant the amount of time that is invested.

I realize that a certain amount of frustration is necessary for learning and this can be a large amount of frustration at times. Maybe the problem doesn't lie within the system but is a basic problem of learning. Maybe we have to be greatly frustrated before we really learn anything.

One possible solution I mentioned to you in one of our discussions. This was to o ffer a problem and i f i t weren't solved in a reasonable amount of time to present it as you would in a lecture. As you pointed our, this approach might be self-d efeatin g . The students might give up on a p a rticu la r problem more quickly knowing that you w ill present i t .

I'm not sure I have any real solution. We as students should prob­ ably put more e ffo rt into the course because this type of course demands much more of the student than lecture courses. Aside from this I have no other concrete answers.

Another, somewhat unrelated point concerns me. This is the amount of class p a rticip a tio n . At times the class drags on with none able to solve a particular problem. In this situation, many times I'll volun­ teer to work on a problem even though I have only a p a rtia l solution or have gained some insight a few minutes before. Sometimes the results are wrong because in my effort to fill in the gaps, I overlook a minor d e ta il. Sometimes I'm not sure i f I should do this as much or if I should hold back on the problem u n til my results were more polished.

November 2k

My views about the course have not changed s ig n ific a n tly since the time of the midsemester interview and writing of the last essay. I do 152 feel now that I am gaining a greater perspective now for viewing the course. I can see a greater unity in the concepts. For example, the definitions of continuity and slope, although they assume some basic ideas such as betweeness and lines, are definitions that rely on pre­ viously developed ideas as little as possible. I guess that I'm just repeating myself again. But again I emphasize the value in taking mathematics down to its barest structure and building on i t to achieve a consistent system. This approach is not only interesting but is also stimulating. It requires the student to think of familiar ideas often in unfamiliar ways. Here again your definition of slope is a good example. Everyone in the course has worked with slopes before, but I doubt i f any of us has worked with slope defined in your context . I think that doing this should help the student to learn to approach a problem from several directions by himself.

Another thing that I've noticed more lately is that in some of the propositions the conclusion is extremely obvious. In these instances it can be very difficult to devise a proof because the student will assume too much. We should always guard against th is . Otherwise proofs w ill become very sloppy.

December 11

Now that this course is almost finished I feel a real sense of accomplishment in relatio n to i t . You proposed two objectives in the introduction to the course. F irs t, you expected us to do some mathe­ matics. I'v e never had a course of this type before and I feel now that more of my courses should have been done this way. One method of teaching says that the amount a student learns is proportional to the amount of involvement the student has with the subject. I feel that the involvement here by actually proving theorems myself has helped me to learn much more than I would have in a lecture course covering the same material. Certainly, we did not cover a vast quantity of material but the methods we used to attack this material should make us more able to go out and learn on our own without any help from a professor.

This brings me to your second objective, that of learning some real analysis. What I said above will apply to this objective also. We may not have learned a large number of "facts" about analysis but we have gotten into analysis ourselves and viewed it from within. For these reasons, what we have done has been much more valuable to me than s ittin g in lectures and reciting ideas that are not developed by us but given to us by a professor. There is a need for lecture courses, don't get me wrong, but this course has been extremely valuable. For these reasons I feel that this course in real analysis is probably one of the best that I have ever taken. 153

Student Six

September 25

The system seems to be building larger and larger, a ll on its past theorems. We are starting to deal in things with which I havent't used in some time, such as in the properties of order and absolute values.

November 11

The proofs I have are not as detailed as they should be. I am having more difficulty as the course goes along.

Student Seven

September 13

I thought you would be interested in an essay this early in the year expressing my feelings before we really dig into the material. As an initial feeling, I am rather apprehensive about the structure of this course. My creativity has never been tested as it will be here, and rote memory w ill be taken from its usual primary position and moved to a distant second position in importance. I guess its this lack of being tested that makes me question the ability of my creativity to measure up in this course.

The lack of emphasis on quizzes, tests, etc and grades in general motivates me to concentrate on working with the material given without trying to memorize solutions, axioms, etc. I feel good about being able to attempt some creative mathematics without fear of being quizzed the next day on some unrelated material. I feel the structure of this course will take the pressure off me and allow me to concentrate on some real mathematics.

October 15

Here it is, one month a fte r my f i r s t essay and my feelings are radically different.

For the last week or so analysis has been very frustrating— I can't seem to get the "big picture" in my proofs and at times make too hasty decisions (i.e. readily extend finite cases to infinite cases). Even when some of the proofs are being given I have trouble "seeing" the rationale used. In the Heine Borel Theorem, the use of S “ (x | [a,x] is covered by a f in ite subset of G} is something I never would have thought of using. 15**

Possibly my problem, or one of my problems is a lack of experience and being accustomed to having a proof laid out in front of me. Possibly it is always having been able to use direct methods to achieve the proof.

Through this frustration, I'm starting to question my ability to do graduate work in mathematics.

Possibly, i f I analyze a given problem f i r s t , re fle c t on the methods we've used fo r proof and don't get hung up on brute force proofs, my luck with the problems w ill improve.

December 18

Although I don't feel strongly at the moment, 1 know you want another essay— so here goes.

Looking back on the course I wonder whether we learned as much real analysis as people would in a course with a tra d itio n a l structure. Possibly we've learned, or have been guided in the direction of learn­ ing, something more important than statement of theorems. This some­ thing is the a b ility to work with mathematical concepts through the form of proof.

I feel too close to the course to really evaluate it. How­ ever, one thing that comes to mind is that too often I felt we were really dragging our feet in class. If more people would do more work, possibly it would have not had these less interesting sessions.

Student Eight

September 27

My in it ia l reaction to this course is mixed. I lik e the course because I've realized that we are in the process of building a system-- a fundamental part of mathematics. Opposed to this is the bitter frus­ tration involved— I guess that, too is a part of mathematics. This 1.17 is an example of a problem that has beenfru stratin g to the whole class. All that we've done so far is simple and basic. Some of what we've encountered is impossible or is possible only under certain conditions. Now for the first time I've come face to face with what a logical system is by actually building one step-by-step. From the "Foundations," the definitions, undefined terms and axioms, we build a complete system. We can see by a lte rin g the foundations we can develop a different system with different characteristics. This accounts fo r the freedom and variety of mathematics.

November 23

I'm getting bogged down I'm afraid. Not that I'm over my head, but ju st that there seems to be a catch to some of these ques­ tions. I'm considering taking **0*t next semester, but I hear that the course may move much faster and I'm afraid I may get left behind. I 155 have to work slowly and deliberately In this course, and if I do that then I usually come up with results. I can see what "rigor" means— at least partially. It takes a lot of concentration and detail to evade the loopholes that can be shot into a proof.

December 16

Well, the semester has drawn to a close and 1 believe I've learned a lo t. I'v e learned something about mathematics from two angles: The structural angle in 403, and the philosophical angle in 451* I won't be taking 404 next semester, upon your advice, but I would be looking forward to other courses that followed the in fo r­ mal style as encountered in 403. I only hope that my grade in this course reflects the amount of effort that I have put forth from day to day. I imagine that its hard to put a le tte r grade on something lik e th is ; I maintain that possibly the pass-fail system would work better for this course. But no matter what the format on grades, I believe that the course has been worthwhile and recommend its cont- i nuance. APPENDIX G

INTERVIEWS

156 TRANSCRIPTS OF INTERVIEWS

Partial transcriptions of the interviews are given in this appendix. Portions of the discussions considered irrelevant to this study were ommited although a few parenthetical comments remain.

Also necessarily omitted are the pauses, tone of voice, and expres­ sions that are so much a part of a live interview and frequently convey more information than the words spoken.

The format of the transcriptions is this: The interviewer's question is stated followed by numbered responses. The numbers correspond to the same students as in the body of this dissertation.

Missing numbers indicate that either the student did not respond or was not asked that particular question. Frequently a student's response led to another individual question and response. This is given immediately after the student's response to the original question and is indented to set it off from the more general questions.

Midterm Interview

What are your vocational objectives?

1. I p]an to teach mathematics in the secondary schools.

2. I plan to go into the army (ROTC), then to graduate school. I want to work in government or industry as a mathematician.

3. Graduate school in economics. I w i11 probably get my masters, then go into the service for four years. If I like the service I may make it a career; otherwise, I will probably go to work in business. k. It depends on the operations research course in the January term. If I am interested in it, I may do graduate work in operations research or s ta tis tic s , then work in government or business. 159 5. Graduate school to do work in topology or algebra. I w i11 possibly teach in college or work in business or industry.

6. It has little to do with mathematics. I am going into accounting. I plan to take courses in Baltimore to take the C. P. A. exam. Our fam ily has a small firm that I am going to work for.

Why did you major in mathematics?

I lik e math and might lik e to teach i t sometimes. I d id n 't go education because there is not enough money in teaching to make i t a career.

7. I want to get a Ph.D. in computer science and numerical analysis, and maybe do some post-doctoral work. Then I would like to work for a research laboratory or a university. I am not really inter­ ested in teaching or business.

8. I worked in V itrio l laboratories during the last few summers and was getting into some computer s tu ff. I want to go into computer work or possibly operations research, if I am successful in the January course. I w ill possibly attend night school while in service and later go to graduate school in operations research.

Why are you taking this course?

1. Because Dr. Lightner told me I had to have i t .

2. Because of the way you are teaching it. You told me last year i t wouId be this way.

3. Because Lightner said I should—for your survey.

4. It is necessary fo r graduate school. It provides background for working with mathematics. Plus, I have to take it.

5. It is relevant to the subject of mathematics. It is needed for graduate school. And, it is of interest to me as a mathematics major.

7. Because it is a mathematics course, it is available, and I like math and wanted to see what i t was lik e . It 's a grass roots course It gets at foundations of ideas and gives a firm foundation you can s ta rt working from. Lots of courses are halfway. You are basing your thinking on assumptions or something somebody says you can do. Here you re a lly s ta rt to see why you can do some things.

8. [Another student] recommended it from taking it last year. I looked at Rudin and found it interesting. Also, it is required. 160

Where does this course f i t in with regard to your vocational objectives?

1. It will be of use in teaching. It has lots of fundamentals.

2. It makes you think and learn methods of attacking various problems.

3. I have no idea. k. It is more mathematics.

5* It is preparing me for graduate work. Itbroadens my understanding of mathematics.

6. Not very much. But building of math systems isquite interesting.

7. It is somewhat foundational for analysis.

8. It is sim ilar to something you do with computers. You are building a system much as in programming. I t provides more insight into what a proof is and what a proof is not. There is more rig o r— sometimes impossible rigor.

What is rigor?

A process where everything is lo g ically derived from every­ thing before it. All the premises are valid statements that you have gathered from what is given or that you have already proved.

Where does this course f i t in with regard to your total undergraduate program?

1. I wish I had had it earlier, before some of the others. It would have been a good background course.

Do you think, fo r example, i t could be taken by sophomores?

I think so. There is n 't enough to tie the math courses together. Maybe this course would help.

2. It combines different aspects of mathematics--group theory, graphs. Graphs are something d iffe re n t here— I never had any­ thing like this before.

3. It is required for a mathematics degree. You get to see different methods of proof. k. You are able to apply yourself to the mathematics as you go along. I

161

5. It is one of several major areas.

6. Somewhat the same. We are building foundations and proofs.

7. it is kind of a continuation of other courses. But it is kind of like a backward thing. You have assumed things a ll along, and here you s ta rt to find out why you are assuming things.

8. There is similarity between this and abstract algebra and analytic geometry.

Do you prefer abstract mathematics or concrete mathematics?

Abstract mathematics is more in terestin g , but concrete mathe­ matics is probably what I will be doing.

Do you like the course?

1. I like it. Especially the informal nature and working things our own way.

2. I enjoy it.

3. Ail right— it's good; not outstanding, but good. I like the format. At first I was skeptical, but I would rather have it this way than listening to you ta lk . I like i t b etter than algebra.

k. I lik e i t fo r as much as I can do, although i t has drawbacks fo r things I am not able to do.

5. I lik e the basic format a lot and seeing several approaches and viewpoints. You get to see the working of somebody's mind as he goes about doing a problem— the thinking behind a proof rather than just a proof. The informal nature is good—sort of like a bull session.

6. Pretty much. It seems to fit very well in the way it is being taught. In a small class and the informal way of the course, I feel a bit freer to ask questions and a bit freer to stick my neck out.

7. I re ally was gung-ho a t f i r s t . It 's not that I don't like the course, but i t is re a lly fru stratin g sometimes. In some of these proofs all the little pieces seem to fit together— it is like a little chain up on the board, this one goes to that one and so on— but you stand back and look at i t and i t doesn't seem possible to go from the first to the last. The little pieces fit together but the whole thing doesn't. 162

What are some specific things that you like?

1. It is 1 ike a calculus course I had in high school.

2. Its relatio n to my paper on in fin ity .

3. Nothing sp ecific. k. Can't say.

6. The informal, small class. It is about as large as you could get for teaching this way.

7. I enjoy the challenge, but feel much frustration. Frustration is part of a challente though. You have to be frustrated before a challenge is worthwhile. Maybe is is just like an interim period. Maybe things will begin to look up and I wi11 really understand things better because I have had to struggle with them.

8. The small class. It forces you to put a lot of time in because you know you are going to be called on sometime during the period. You must have at least one problem done as well as you can. There is lots of pressure and incentive to put a lot of time in on the course. I also like the thing about getting away from the te x t­ book. This is something that you are building from simple rela­ tions.

What are some specific things that you dislike?

1. I dislike the fact that I can't get many of the problems. I don't have much of a feeling of reward or anything. I feel my work is fru i tle s s .

2. I d is lik e being frustrated on a problem that I work on many times but ju st can't get.

3. It seems like a waste of time when some people just stand at the board.

4. Not having done things. It is generally interesting, but I don't see anything that stands out as a 1 ike or d is lik e . I would like to see more use for some of the material.

5. Dragging our feet. We are not covering that much real analysis. There ought be be some way to move fa s te r. Maybe more work should be put in by the students to speed up. Sometimes I drag my fe e t. I work on a problem and get hung up on i t . Eventually I ju s t give up. This is the kind of course where you have to have the students working hard. It is almost completely depend­ ent on the students. 163

6. In some cases I would like to have examples. I sometimes can't figure out examples on my own.

7. Being frustrated and not seeing some things that seem obvious later. It is not really a dislike, but a frustration.

8. Can't say I dislike being frustrated. At the time it is very unpleasant, but when I see i t on the board i t is so easy.

How can I improve the course?

1. A couple of lectures here and there would help. You could work some of the more difficult things and explain some of the defini­ tions .

2. Leave i t the way i t is.

3. It is good practice for people to criticize other's proof. k. Once in a while the instructor should show uses for the material. Explain the relevance of the problems other than ju s t to prove other problems. There is a need for application of the material to other areas. For example, how could it be used to solve prob­ lems encountered by an engineer.

6. By putting in some more examples; however, too many examples could take away from the work we do.

7. Sometimes I wish i t went fa s te r— sometimes i t drags. Sometimes it might be good for the class to work together.

How can I improve the course fo r the rest of the semester? k. I need more help for some of the problems.

How do you want me to do kOkl

Individual meetings would be h elp fu l. I f I keep not getting proofs, this kind of class would not be very helpful — lik e now, I'm ju s t going there and I see this guy doing a proof and whoopee. A book would help out.

5. Itls going to have to be the student who wants to work harder. Maybe allotting certain periods of time, then the instructor making more e x p lic it hints or working the problem would help. It is d iffic u lt, once you have been frustrated on a problem several times, to go back and work on it with much enthusiasm.

7. I wonder if we learn as much real analysis as other people. 164

8. I can't see anything I would like to see changed without changing the basic outlook of the course.

How can I improve the course for next year?

1. Do i t s im ila rly , but more lectures and examples.

How much would class size a ffe c t a course taught this way?

A lot, I don't think you could have a class this way much larger than ours is now. I would say that i t wouldn't work for over ten people. In fa c t, in our class [two students] get most of the solutions with sometimes [two other students]. In a large class, lik e over ten, I think it would be the same people doing all the presentations and after a while i t would get to where the rest would ju s t s it back and watch.

2. I d e fin ite ly recommend that you do i t this way.

How do you want me to do 404?

Do i t the same way. This is one of my most interesting courses that I am taking this year.

3. Do the same thing over again. You could perhaps put more in i t . There are places where we need more lemmas for theorems, etc.

4. Do the early development s im ila rly , but as the m aterial gets more sophisticated make use of a book as reference.

5. You should keep basically the same kind of format. There should be more hints on the study guide for difficult problems. On particularly sticky things go to lecture for one or two periods.

How do you want me to do 404?

Definitely keep it seminar type.

6. Continue the method. Where students have to work out proofs themselves, it has to be informal so the students w ill correct themselves. In the informal class, there is more room for com­ ments from others. There seems to be more eagerness to under­ stand.

7. It depends on the outcome this year. If there is a decent correlation between what we know and understand, and what others would, then run it this way. I think this way is preparing us for graduate work. 165 8. I don't think it would be a bad idea to continue the course as is. I would object to doing exercises in the book. I can see using the book fo r reference about specialized aspects such as Dedekind cuts or Cauchy sequences, but can't see it as a basic reference fo r the course.

Do you like the teaching method?

1. I like the method. I like anything that is more individualized rather than just lecture.

Do you anticipate using any of the techniques you have experienced here?

I would lik e to. I think i t is a more d if f ic u lt technique than straig h t lecture, because this way you are subject to more questions. We have more time to think about the material and bring up questions. Sometimes I think that you must think we are wasting time— like when we are presenting a wrong proof.

Do you thing we are wasting time?

No. I think we are learning as much from the wrong proofs as from the right ones.

3. In this course i t works very w e ll. I t might not work as well in a course lik e algebra.

4. I generally like i t . It is better fo r some than others. It is very well for some.

5. Overall I like it—more than the method used in 317 or probability or any other course I have. At least while I am in class I am always pretty much a le rt as to what is going on and I try to keep in with i t . I fin d i t sort of exciting in a way. Some of the other courses at times get sort of lackluster and my mind wanders.

6. I like it a lot, very much.

7. Its frustrating. I like it. Maybe the proofs will help break some chains. After three years of college math, you are just used to opening the book and studying and spitting the same stuff out again or in a slightly different form. Maybe this will make us really start to think about the things that we do. I am becoming more aware. I am questioning more in the other courses. I think I am more perceptive now. I am forced, instead of just studying a proof and following somebody else's reasoning, to make up my own reasoning.

8. It puts pressure on the student. He knows that he is, in effect., 166

going to be teaching the class and must understand the proof well enough to not only understand it himself, but to explain it to others.

Is the approach d iffe re n t from what you had previously experienced?

1. There was the high school course mentioned previously. In a way, it is like an English criticism course.

2. D e fin ite ly . You touched a l i t t l e b it on i t in abstract algebra when you gave us a hand-out to develop ourselves. But I had never had a whole course this way. Projective geometry is close.

4. Yes, def ini tely.

5. I had something like this in high school general physics. I liked i t there.

6. Yes. They have been instances in classes where such things happened.

7. Radically d iffe re n t. Most courses are lik e calculus. That is you cover a set amount of material in class, study it, have quizzes and tests. The stuff you have to work with at home is maybe slightly d iffe re n t from the s tu ff you do in class, but you use the same basic forms.

8. I t is sim ilar to the Math 451 seminar. I had a high school course called critical thinking. The teacher set in the back of the room and bounced questions off the blackboard for us to react to. I learned a heck of a lo t that way.

I f you had known that the course was going to be taught in this manner, would you have enrolled in it?

1. I think so.

2. I did know it was going to be taught this way.

3. When I was buying books, I saw that you had a book recommended but not required. I thought how in the world can you do a course with no books. Yes, I would take i t again i f I knew ahead.

How would you lik e to see 404 done?

Continue this way. I don't like to read books.

4. I would probably have hesitated, but might still have taken it. I didn't expect the course to be this different. 167

5* Yes, I sure would have, especially a fte r that course in high school. I would take about any course I could get my hands on that was seminar oriented rather than lecture.

6. Very d e fin ite ly .

7. Yes.

8. I would have probably taken it even if it had not been required.

What have you learned?

1. A lot is familiar, but was accepted without proof previously and had been forgotten.

2. Quite a bit, especially about graphs. You learn more by trying to do i t , even i f you don't get anywhere. I f you try three or four different methods and don't get a solution you s till under­ stand better about what you mean by the definitions and stuff lik e that. You get good ideas from class many times— things you didn't think of.

Do you like a small class better?

Definitely, much better. I think in a small class, opinion can be brought out much easier, whereas in a big class not. You can more or less associate with one another and become friends throughout the semester.

3. I have learned how hard i t is to prove some of the things we take fo r granted— lik e the existence of the square root of two. Also terms like cluster point, covering of an interval, etc.

k. I have a strengthened basis fo r knowledge of real numbers and some new ideas about math.

5. Not so much real analysis as I might like to have learned. I think there are valuable things besides this though, like approaches to a proof and ways of thinking about something— different attitudes toward working a problem. I might be developing the a b ility to in tu itiv e ly work with something and from that intuition to construct a proof or work the problem, whatever is needed. I t sort of gives you a base of ideas and knowledge needed to work out a problem— a much broader base than you would get by just learning definitions, theorems and proofs.

6. To be more rigorous in proofs.

7. What you c all mathematics. I have learned to work and see relationships I d id n 't see before. Maybe I have become more 168

creative. I don't know. I hope so. There are few new rules, etc. I have learned a method of thinking rather than material.

8. The rigor of proof. I have a different and better outlook of what a proof is. I am learning about mathematical systems--what they are and how they are derived.

Do you feel that you are learning more or less than in a more con­ ventional lecture-discussion course?

1. I don't know. I can see the advantageof having tests for the review necessitated. I haven't reallysat down andreviewed the m aterial. I think we have a tendencyto forget about e a rlie r material and not use i t in proof .

2. I am learning more about doing proofs and seeing why or how things happen. In lecture, you might learn more in quantity, but would not re a lly understand why i t works or how it works or where you got some proof.

3. I like to think I am learning more. You learn it, if you do it yourself. If you just write it on the board, we don't learn it.

4. Less, we are not getting as much done.

6. More, I feel more motivated to work.

7. It depends on what you call learning. If learning means thinking for yourself, we are really learning something. If it is just memorizing a bunch of rules, no we are not. We could have learned a lot more i f we ju s t had the book in front of us and memorized Rud i n.

8. In lecture courses, I have a tendency to miss assignments. There is not as much pressure to get in there and spend some time. I am working almost everyday on this course, because I know I am going to be called on.

Same question with respect to real analysis.

1. Probably more—whatever real analysis is.

2. We are probably not learning as much subject matter. But I think that the fact that we are doing it ourselves makes up for that fact—overshadows that fact.

Do you think you could pick up Rudin now and read it meaningfully?

Yeah, more meaningfully than i f you took the course and ju st read through i t . 169 k. We are going slower, so we are probably learning less.

5. There are two ways to look at i t . Like exposure to m a te ria l--! haven't been exposed to as much different material. On the other hand for learning insights, attitudes and approaches, I have probably learned a lot more. Maybe this is more valuable than ju st exposure to m aterial.

8. I have a tendency to accept something as true without thinking about i t in a lecture. I have had to do a lot of thinking about these problems— the question of "what if?" must be considered.

What is real analysis?

Working with real numbers. It is mainly proving things from given relationships, and taking these properties and using them in such a way as to build a system that has ome p ra c ti­ cal vaule.

Same question with respect to mathematics.

1. Hard to say; probably more.

2. You are learning more because you have to formulate these things yourself, whereas, in lecture you see the professor do it and say "okay, I believe it." I think you are learning more by doing it yourself. The methods you use like proofs are relevant throughout mathematics. k. You would learn more following Rudin; although you might not have as much understanding with the teacher doing the proofs instead o f doing them yourself.

Are you spending more or less time on this course than you would in a more conventional course?

More. You try to work things out yourself. If I see you put something on the board and I understand i t , I won't look at it again until it is time to study for the next test.

5. More in this realm.

6. I am learning more about proofs and mathematical principles.

7. How do you define mathematics? Most people think mathematicians just learn a bunch of rules. I think we are learning more mathe­ matics. I just wonder if we are able to think more creative now than if we had ju st studied Rudin. 170

8. Yes, that is perhaps why it was chosen as a required course for math majors. Rigor of proofs and construction of systems has a lot to do with more generalized mathematics. This seems to be more basic than Rudin— more general mathematics.

Same question with respect to teaching-learning.

1. Definitely more. The advantage of this set up is that we learn things more thoroughly as we go along, rather than letting it siide unti1 a test.

2. You learn by doing.

3. I think it would be a good experience for the education people. I am beginning to see more and more that you learn by doing.

5. I have learned that learning can be awful frustrating. Once you have got the s tu ff i t is good.

6. You are learning by doing.

7. Guidance is helpful.

8. You have to learn the material first, then study it and become adapt enough to teach i t . It takes a l i t t l e more than in tu itio n .

How do you rate (grade) yourself fo r the course to this date?

1. I don't know. I have put more e ffo rt into i t than I have shown results.

Are you spending more or less time than usual on this course?

About the same amount. It depends on my interest in a p a rti­ cular problem. I may get frustrated and give up.

What do you do when you get frustrated and give up on a problem?

I always try to do my analysis f i r s t for the following day. If I can't get anything, I sometimes do something else and then go back to i t . Sometimes I can see i t b etter that way. The reason I do it early is because I think i t is a course where if I didn't want to do anything then I really wouldn't have to, and I am afraid that if I left it till the end I might not do anything.

2. How about a pass?

3. Good—not outstanding, but as good as everybody else. I contribute as much as anybody.

h. About average; C maybe. 171

5- Probably a B.

7. B. I haven't done much work. I try and i t upsets me, but I just can't do It. It hurts to know that I am trying like hell, but ju st can't get any work done.

8. Some of the m aterial is way above my head. I guess I would give myself a C. I don't think anybody is failing or getting a D. I think it should be pass-fail rather than letters. It seems unfair to draw lines like that.

Do you have anything that you would like to "get o ff your mind" at this time?

1. For some reason, I can 't see through a proof. I can see i t a fte r i t is presented, but I ju st can't think through i t .

2. I am having a h o rrible time trying to prove the existence of the square root of two. Almost every night, I look at that and think there's got to be a way. This course brings in a lot of other parts of mathematics. Frustration gets me a lot of times.

8. I think the grading system should be changed.

Do you think I should give some tests?

That would be almost contradictory to the course. I look at everyday as being an exam. You can gather enough evidence from day to day to determine whether the person is doing satisfactory work.

Do you think you are spending more time or less time in this course than you would in a more conventional course?

I am d e fin ite ly spending more time on this course than in lec­ ture courses— sometimes a riduculous amount of time. No doubt about i t .

Is the time spent worthwhile?

It is worthwhile i f I prove something; it 's not worthwhile if I don't prove something. However, even i f working in circles you learn more by t r ia l and e rro r, and e rro r. You have a bet­ te r Insight into i t when you see somebody do i t and see that it actually can be done. I haven't been exposed much to proof by contradiction. I have had to do it more in this course. It 's hard fo r me. 172

Final Interview

What is your overall reaction to the course?

1. I think I started out being very pleased with the course, then I went into a slump, and then came up at the end. It wasn't the course so much as i t was me. I think I could have had a better attitude in the middle— I just gave up since I couldn't understand some of the things in there. I liked the course. I think I probably enjoyed that math course more than any other I have taken up here because I d id n 't feel so pressured.

2. I enjoyed the course—probably as much or more than any other mathematics course. This was the f ir s t seminar course I ever had. I t was d iffe re n t to me—you d id n 't know what to expect— at first at least. It turned out to be quite an experience.

3. I don't like that. Skip that one. You are trying to get an overreaction. It wasn't that exciting. No, I thought it was pretty good. k. For me i t d id n 't work out so w e ll. It was the f i r s t mathematics course I ever had much trouble with at all. I think it is a good way fo r some. I know others liked this course, perhaps because they are more interested in pure mathematics than I. I am more interested in taking what you have and working things with it . I think that is partly what the trouble was.

Do you think the main problem was in motivation?

I think that was some o f i t . But I do have trouble with proofs and I always have. I d id n 't do well in the course, but I f e lt like I knew what was going on. I t was ju st doing it for the first time before someone else put it on the board. At first when I started having trouble I thought i t would work out, but I ju s t kept getting worse.

Do you think some applications could be built into the course with this teaching method?

I don't see how. I t would help me i f it could.

Do you think you would have done b etter in real analysis with a more conventional format? 173

I don't doubt that at a l l . I would have been much more accustomed to that. I generally do take time to adjust. I know that back in high school I always had a bad f ir s t term before I finally got back in the groove. Alsc* in calculus I had Dr. Spicer and we didn't do any proofs in there. When we got to the calculus part I hadn't even worked with them at all there and that probably hurt too. We ju st worked out problems, which suited me but I missed out on that.

5. I t was probably one of the best courses I'v e ever had. It 's taught me to rely on myself rather than sit in class and just take everything. To prove something, I have to s it down and figure all I need to prove, how I'm going to go about it, figure out a way to w rite the proof, then check i t to make sure I'v e done what I wanted to and not made any logical errors. It is sort of inward depending rather than depending on some­ body for results. I think that is about the best way to learn mathemati cs.

6. I thought the course was good, in terestin g , because i t set up a system and built the proofs that you hadpreviously proved. You keep using what you had in the new theorems that you came up w ith. It gave me a good sense of what proofs were about.

7. Basically a good reaction. I enjoyed the freedom in the course. I enjoyed the lack of necessity to cram. I just liked the set up where we could work on problems at more or less our own pace as long as we kept up with the course. I didn't find keeping up with the class difficult really. You could sit down for three or four hours a couple of days in a row and whip out a whole mess of problems. Of course there were some you would miss and have to go back and look a t. O verall, I think I have a good reaction to this set up and the way it was structured.

What do you think could have been done to speed up the class?

You could have been more selective in the students you let come into the class. In overall terms no matter what the students, I don't know how you can get them going. Maybe we dwelled on some of the problems too long or could have skipped some.

8. I like the course. It was like a refreshing change from some of the other book courses. I think that i would have had a lo t more trouble getting along in a book course than in this course. There was freedom in i t . I don't think i t was radical. I had a feeling of doing something. I got a feeling of structure and the structures of mathematics—the scale, order, patterns, foundations of mathematics. Other courses have talked about these things, but we never did do any mathematics. m

Do you have comments on some specific instances from the course?

1. I really liked the way you handled it. It was very indivi­ dualized and very personal. You had more opportunity to learn at your own rate than in any other math course that I have taken. I know one thing— I d id n 't memorize like I do in a lot of math courses. I'll admit that in other courses, I sit down right before tests and memorize proofs.

2. What I especially liked about the course was that when something came up you didn't just step in, but said okay if that is accept­ able to everyone. There were more than one way of doing things and I liked to see more than one way. I thought that was quite interesting to see how different people think.

3. I can't think of anything specific worth commenting on.

4. Nothing in particular that much.

5. Just the whole approach is so much closer to re ally what I think education should be. It may not be perfect or the final answer. But i t seems lik e too much of i t is just memorizing facts or memorizing theorems and repeating them again for a test. This way you are actually learning to do rather than see things done for you.

6. I thought the times when someone would s ta rt a proof and get stuck and you would le t him work more on i t for next time were good. That way he made something for himself.

7. I was ju st thinking of some of the problems I worked near the end of the course and spent so much time on and one was wrong.

8. The proof that one was a positive number— you are assuming things that you think are so obvious. The book says one is positive. Why is it positive? I didn't really think about things like that before, it was thought provoking. I thought i t was a real interesting course.

What part of the subject matter did you like the most? least?

1. I d id n 't lik e sequences as much because I couldn't understand them as well. The parts I didn't like were the parts I couldn't understand, really.

2. I liked the beginning part best. Properties of real numbers— I knew most of that stuff. Continuity the least, especially that d e fin itio n . I really found trouble trying at times to work with i t . 175

Do you think you could work better with just an e-6 definition?

I think so, because it 's there and you can apply i t . The other has a ll those vertical lines and so on which kind of threw me. A lot of those definitions, I had trouble trying to understand. I think maybe more illu s tra tio n s would help.

Do you think they should be b u ilt in as a part of the course or given by students or what?

Maybe in certain cases where it is definitely necessary they could be built in. But in other cases if you had maybe someone in class to give examples then i t might be better. For then the student expresses it in his own words and the other students might catch it better.

3. I liked the part that wasn't very topologically oriented. I liked the beginning p art. That was good as fa r as interest because it wasn't so abstract like sequences and nested intervals and all that stuff. I like things that you can see. Ialso liked the parts where you gave examples. Like the problems on property (C ). That was good, but that was about the last time you had any kind of concrete examples.

Do you think these examples should be b u ilt into the notes?

Yes. Sometimes you said give examples of ___ , and that's a ll right sometimes, but I would like for you to put in more . I liked the part at the beginning where you worked more with numbers and less with theory. k. I guess I mostly liked working around with problems on property (C) and slope. I had some trouble there, but it was kind of interesting how we worked around with them and some of the things you can show with them. The least was sequences. You d id n 't have that much in there on them and I d id n 't see a whole lot to them. I think for the other there was more to it there and that helped some. I thought the f ir s t part was interesting— proving those l i t t l e things you always knew, but you never had proved before.

5. I liked the topological properties and metric spaces the best. I liked the very beginning the least. It seemed so basic. I f e lt lik e where are we going? Once we got there I saw where.

6. I liked the first part best, because I understood it best and I enjoyed working on i t , using what I know and what is laid down on the sheets. I t was through absolute values that I liked best. What I liked least was the part where it began with slopes.

7. I certainly didn't enjoy the beginning. It is hard to say though, because that was when I was being introduced to the 176

course so that may have had some e ffe c t. It might have been my initial reaction to getting all these things to prove. Anyway, I wasn't re ally excited about the f ir s t part. I think I got a kick out of the area around the Bolzano-Weierstrass theorem. I could see topology coming in here and I didn't feel so lost; that was the thing. It was a new feeling. I said, "hey, I can see some of this now and I understand some things."

8. I guess what really got my interest was the f i r s t few pages. Every­ thing looked so obvious, but I found you had to really give i t more thought. The f ir s t problem I presented in class was an addition problem, but I d id n 't take into cccount that some things weren't defined and that you had to list every single step. So in the next problem I presented, I had to think about twice as hard to make sure I got every single detail in, make sure everything was defined, and make sure everything had been given before. I guess that gave me a little bit more insight into what a mathematician is , what he does, what he expects on a problem, and what other people expect of him. I can't think of what I liked the least. I ju s t enjoyed the course. It is one of the few courses I have enjoyed. I can't think of anything I didn't enjoy. I was a little dubious at f ir s t about going to class and talking into a micro­ phone, but I soon forgot about i t .

To what extent did you attempt to recall the proofs of familiar state­ ments from previous courses?

1. Not very much at all. My abstract algera was two years ago, and none of the properties were proved in calculus.

2. At first some, but after seeing the way the proofs fit together, I trie d to look at the other problems to get ideas.

3. I trie d to s ta rt over from the beginning. It d id n 't impress me that there was that much overlap. k. Not too much. I stayed away from books. I d id n 't recall proving many of them before. We didn't prove any of the calculus problems before.

5. I t was not a regular thing. Even theorems that we had seen before were based on d iffe re n t d efinitions so we could not rely on the previous proofs. In the early stages the overlap is necessary. We were basically rethinking things. If we had new material, we would not only be trying to learn how to prove things, but new material too and it might be too difficult or almost impossible when we got bogged down. Later in the course we do th is , but at f i r s t we need to be able to work from a basic knowledge. I don't think for the average student that it was that much familiar. 177

6. In the beginning, i t was mostly something that we had proved before, but not in later proofs. The methods of proof, I did recall to a degree. I found i t important.

7. Where I had seen a thing before I tried to recall the proof. I don't think it really matters whether the material overlaps other courses or not.

How did this course a ffe c t your view of "what is mathematics?"

1. I think it helped me to pull together the different courses. It gave me an overview of mathematics as an area rather than d is tin ct sections.

3. That is a dumb question--you asked me that last time. No, it might have changed a l i t t l e , I don't know. Taking that and algebra at the same time sort o f, maybe both worked together. It seems more abstract. The further I go the more abstract it gets and the less I like it. Like that's real interesting when you can take cost and revenue and subtract and get p r o fit, and then figure what maximum profit comes using calculus. That's very good.

4. I don't think so. I am more interested in using it.

5. I f anything, it make me re alize that most of the courses we take a re n 't re a lly mathematics in a true sense. They are learning about mathematics. They are learning what somebody else has done with mathematics. But they are not actually doing mathematics. I think in this course, we were doing mathematics. I t might not always be original or very high powered, but we are attempting to develop something on our own. For a lo t of us i t was probably the f ir s t time that we really were doing our own mathematics. So in that sense, it affected my view of mathematics. I realized before that mathematics was more than somebody taking a theorem and w riting i t on the board. But, this course made me more aware of what actually doing mathematics is.

6. I don't know. I don't think so.

7. No, because I don't have a firmly established view.

Did your view change?

I don't think really it has. Even before the course, I knew that mathematics wasn't just calculus or algebra or geometry.

8. I got a b etter insight into the way mathematics works and what the foundations of mathematics are. It c la rifie d things that you read about how mathematics builds structures. I re a lly had to work with it, struggle with it, get frustrated with it—if that's what you 178

have to do, then I did i t . Maybe you should get frustrated sometimes.

Do you see mathematics as an activity?

1. No. I see mathematics more as a way of thinking?

2. Yes. I think a ll courses should be like th is .

A. Yes. It can be thought of that way.

7. I kind of see math that way.

What is the role of numerical problem solving, axioms, d efin itio n s, theorems, proof, and counterexamples in mathematics?

1. Problem solving is scientific; the others are artistic.

2. Numerical problem solving is in the part where you want to provide a solution or come up with an answer. You want something tangible at the end. Whereas with theory, you want something where you can begin to sta rt building. In other words, problem solving is a tran sitio n between theory and your answer.

3. Numerical problem solving keeps your interest for one thing. You can use it to figure things out. Plus, it makes it easier to demonstrate some of the more abstract things i f you have examples. It makes i t more meaningful. I guess there must be some use for all this abstract stuff. You have to define where you are going to start and figure it all out. It might teach you methods of thinking. It might help you to think analytically. But, the ultim ate value of mathematics is in its usefullness. k. The theory is necessary to develop ways of solving numerical problems.

5. They all have a particular place of importance. A numerical prob­ lem may not be as valuable to a theoretical mathematician, but it can provide valuable results to the sciences. It can be related to the things around us and be useful. But without the theory, there would be no mathematics—even numerical problems. Structures have to go with the development of mathematics.

6. I think mathematics is a science in which systems are produced; where mostly they are built on foundations. Some systems use the rules that other systems do and some throw some rules out and go on that. Problem solving can, to a degree, verify a theorem, although it still has to be proved. But, all you need is one problem to disprove a theorem. Problem solving is important because the more you use theorems to solve problems, the more familiar you become with these theorems. Axioms are minor proofs in themselves. You distinguish axioms from proofs in that proofs 179

are more general and the axioms are evolved from the proofs--are certain work outs of the larger proofs and the larger theorems.

7. Numerical problem solving is more in applications. Axioms are basic. Theorems are basic too, but more up in the hierarchy. Proofs are a method of establishing theorems.

8. Axioms are guidelines and rules. There are certain important gen­ eralizations we can make from nature. Theorems are basic things that can be proved. You expect nature to abide by theorems.

Did this course affect your a b ility to w rite proofs?

1. It probably did. It helped me to be able to think more logically and deductively. I had never been required to write proofs before. In other courses I had seen proofs, but was not required to do them. The teacher did a ll the proofs and I just memorized them. This was the f ir s t course that I really had to prove something.

2. Yes. I don't know whether it is due to this course or not, but I d e fin ite ly think I can w rite a proof b etter now. I imagine it would have to come from this course.

3. It helped my ability to write proofs, definitely. Before I took the course, I wasn't so indirect proof oriented. It may wear off. Somehow I started thinking that way. If it doesn't look straight­ forward, I try to prove i t backward— that i t can't be any other way.

4. No.

5. I think it has helped me to be a little more clearer with what I am doing. If nothing else, it has helped organize thoughts in the sense that I see a given amount of information and I want to prove something from i t . I guess it 's sort of a systematic approach, you look at what you are given and put the given ideas together in various ways to see what you can develop from them. You see i f that can give you the result you are a fte r. If this is not success­ fu l immediately, maybe you take the result and by some backward process approach some of the given ideas. More than th is , I think I t has ju s t helped me generally with in tu itiv e knowledge, so to speak, as to how to approach a problem—what to look for. It is d if f ic u lt to put your finger on i t and say this is what I learned. It's just a feeling like you know better what to look for.

6. The course helped in w riting proofs, because I never had much experience in writing proofs through high school and college.

7. Between this course and topology, my a b ility was affected.

8. I think i t did, especially a fte r the f i r s t couple of problems I set up in class. I got to be a lot more c r itic a l about some of the things I say. I had to become more c r itic a l and put a lot more 180

time into it. There were lots of times I would sit over lunch and think about some to these problems.

Did this course a ffe c t your understanding of what a proof is?

1. It helped me to understand how you go about proving something. I think I understand more what proof is.

2. Maybe I have changed my way of using proofs, but I don't think it has affected my understanding of what a proof is. k. It helped some on understanding. I saw lots of different ways fo r doing proofs.

5. I feel like I knew what a proof was before the course.

7. Oh yeah. It was good having these two courses [topology and real analysis] together.

8. I think it affected my understanding of what you have to do to prove something. It gave me insight into rigor--what you should do to get a rigorous proof. I can't say that I'm an expert on it, but I have much better insight into what a rigorous proof should have and what you should do to get a rigorous proof.

Did this course a ffe c t your a b ility to read mathematics?

1. I think so. I never trie d to read math books before, because I always f e lt lik e I couldn't follow them. In here, I had to read some of the problems over and over to try and understand them.

2. I think that it has helped. As you read the definitions and axioms you can see what you are getting at as you go to the problems. I think I could pick up an analysis book and read it better now.

3. i don't think so. k. I don't think so.

5. To some extent. We had to read to determine what the definitions meant rather than ju s t have the instructor te ll us what they meant.

6. It has helped me quite a bit. It has acclimated me to the idea of what to expect and just the idea of experiencing thesame set­ ups—not just the presentations, but the ways of proving and disproving.

7. It gave me a l i t t l e broader background. An a b ility to see what someone else is proving, the a b ility to follow i t . I t has widened my a b ility to do a proof, so inthe same lig h t, i t haswidened my ability to understandother's proofs. I can look at the stuff 181

in the calculus I text and I see, not fallacies in the proofs, but where they assume things they shouldn't assume, just to make i t easier fo r calculus I students, and I can see where it is not really valid.

8. I am more conscious of statements that say " i t is obvious" or " i t is easily done" and such cheap words in books. The course has made me a lot more c r itic a l aboutwhat I see. I tend to be more skeptical about statements u n til I see that they have really been proved.

Did this course a ffe c t your creativity?

1. I t did a ffe c t my c re a tiv ity as fa r as proving was concerned. In terms of thinking more creative overall, I don't think so.

2. It forces you to be creative.

3. I think i t helped somewhat. You have to think more, because of the fact that you are dealing with these proofs all the time and these abstract things that you can't even visualize. You have to be more creative to think about d iffe re n t approaches.

4. Maybe a little bit. But I never was very creative. Maybe that was part of the problem.

5. I t depends on what you say is c re a tiv ity i. If you say it is approach ing mathematics and being able to see what is needed to w rite a proof, to see what a d e fin itio n says and be able to explain i t , then I think i t has helped c re a tiv ity to improve i t .

7. I guess i t did. That is a hard thing to answer, because just coming out of that course I haven't had a chance to practice it any place but that course. I guess really it has, because it has given me a chance to exercise i t .

Did it affect your ability to create your own proofs?

5. I t has improved my a b ilit y , but that is maybe overshadowed by the fact that It has improved my confidence of what ability I have. Before, we may have been able to write a proof, but were seldom called upon to do i t . We had never been challenged before, on any great scale, to write proofs.

6. It has helped me in that I didn't have much experience before. I think I am a b it more able to create my own proofs now. The only thing is it would be nice to find out whether it is cor­ rect or whether it is not.

7. Definitely, I am more able to create my own proofs.

8. I think so. Really, I didn't have that much experience with 182

proof. Calculus was a ll problem solving. The only course to require proofs before was 307 , and I didn't exactly ace that course. I think I can do proof now a lot better than I could last year.

Did the course a ffect your confidence in your creative a b ility ?

1. No.

2. I think I am more confident now.

3. I suppose, somewhat. I would be more confident of solving a prob­ lem now. Not because of an increase of knowledge, but ju st because I might be able to attack it from more different ways.

*». I don't have any more confidence in my c re a tiv ity —maybe less.

5. I feel more confident; not necessarily more successful, but more confident that I can approach something, and given a certain amount of time, come up with something that may seem reasonable.

6. I am a bit more confident in proving theorems, because I have had more experience. I've worked them out more than I had before.

7. Yeah, I am more confident.

Did this course affect your attitu d e toward learning mathematics?

I. Yes. I have realized that you can't get by on memorizing.

3. Not too much. h. Probably negative, but I have tried not to let it affect me. Now that the semester is over, I can forget i t .

5. I think at long last I have learned some mathematics, rather than ju st memorizing mathematics.

8. You learn by doing. A lot of courses have theorems stated in the book, but there is no proof and you don't prove them in class. You work practical problems. But I didn't really get that much if an insight into it; into why I was able to do this problem. I d id n 't question the theorems because it was w ritten right there in the book and so was true. Mathematics is something that you learn by doing.

Do you experience a sense of accomplishment when you complete a proof?

1. I experience a feeling of success, and if I don't get it a feeling of frustration.

2. Definitely. 183

5. In a way. A proof is like a puzzle sometimes.You work and work on i t , and fin a lly you conquer i t , and you feel good.

6. I do experience a sense of accomplishment when I complete a proof. However, a fte r the feeling of accomplishment, I have feelings of doubt as to whether i t 's v alid or not. But, yeah I do feel a sense of accomplishment for I feel that I have worked with a system— lik e I have b u ilt something and put a ll the parts in.

7. Yes.

8. If I find out it is right when I present it.

Did presentations with errors discourage you?

Sometimes a little bit, like if I had put a lot of time into the p rob1em.

Did you feel intimidated by your peers when they picked on your proofs?

A little , especially when [a student] did.

Do you feel motivated to learn more real analysis on your own?

1. I am sure I w ill come up on some of these ideas in my teaching.

2. Yes. Plus the fact that when you do some of these proofs, you keep working on others like the chain rule and Cauchy property.

3. I don't think so. I'd rather learn something more concrete. k. I wi11 probably take some more analysis in graduate school. I am taking complex next semester.

5. I feel motivated to. I don't know if I will.

6. I don't know.

7. I think I have been motivated for quite a while to learn math on my own and do so. I wouldn't say just real analysis itself. I kind of try to learn on my own what I enjoy, and i t changes. Sometimes I ' l l do some pro b ab ility like Markov chains or something, and la te r I ' l l look at something else. I'm not too consistent in what I do.

This was a ch aracteristic of yours before the course though?

Yes. The course hasn't motivated me at a ll ju st to do a lot o f extra work in real analysis.

8. I wanted to take kOk, but you advised me not to. 18*t

Do you think that I should use this teaching method in my other courses? which one?

1. I think you would have to have a small class for i t . I can't think i t would work in calculus. I can't really see any other courses, i t would work fo r.

Why would i t not work in other courses?

I wouldn't want to take a course like this as a sophomore. I don't think the subject matter of other courses like cal­ culus or algebra or geometry would lend it s e lf to this method of teaching. I don't think you could get by without having some lectures or some kind of instruction. When we took this course, we already had these basics from other courses. I don't think you could expect students to learn these on the i r own.

What would happen with a course lik e this in terms of subject matter and teaching method for sophomores?

A course lik e this on the sophomore level would change the student's whole way of looking at mathematics from the begin­ ning. I t would be good, but I don't know if they would have the basics to rely on.

Do you think they would be motivated enough?

I don't know. It might be so much over th e ir heads that they would ju st give up.

2. I think a ll courses should be lik e th is . No, you can't teach a ll courses lik e this but courses where you can build from axioms such as geometry should be done this way. I think it would be hard to do calculus this way.

3. I think i t would be sort of boring i f you had two courses like this a t the same time, or i f you had fiv e semesters of i t in a row. That might be too much. Maybe i t wouldn't. You might get used to it and accept it. It is sort of a painless way, especially with no tests. You probably learn less material but more about methods of proof. I think i t would be good for the earlier courses. Then the student could use the proof tech­ niques in his la te r courses. k. Not e n tire ly . Maybe in some parts of the course, but not through the whole course.

5. I don't know if you should use It in all your courses. I definitely don't think you should use i t in freshman courses. I don't think they are prepared for something like that. The shock might be too great. 185

6. If its practical. It depends on if things have to be presented by the instructor for the student to understand or whether he can work i t out himself from the h in ts, etc.

7. No. I don't think i t would worktoo well in something lik e cal­ culus or required courses. I don't know if people would be moti­ vated enough to do the work. From what I've heard, they had enough trouble just getting homework done and things like that. I don't know. I think i t 's a good method and has a lot of poten­ tial if it's not misused by students who just sit back and let the others do the work.

8. Yes. I was happy to find out that you are going to be using it to a certain extent in complex. I think you should. I think it is made for a small class. It would probably be impractical for freshmen. It should be more applicable for junior and senior level.

What modifications in the teaching method would you recommend?

1. For this course, I prefer this method to lecture. Maybe there should be some way to make sure they understand the problems, perhaps a midterm exam.

2. Aside from more illustrations, I can't think of anything.

3. I don't know of any. I t would be good i f you could get everybody to participate equally.

5. Provide some background on d if f ic u lt problems where everyone gets bogged down. Develop the students' feeling for i t so they can work it on th e ir own.

6. You have to have a small class. Ours was about as large as you could have for i t to be e ffe c tiv e .

7. I really don't know how you can get it going faster. You could have conserved time in class by having more than one proof w ritten on the board at a time in some cases.

8. More extra credit work—a couple of things to hand in. Grade as pass-fail. Use a text at some points—do the sheets, then work in te x t.

Should students be allowed to work together? use texts?

1. A fter the f i r s t month or so on some of the more d if f ic u lt theorems working together might help. That way everybody could contribute their ideas.

2. No. I don't think they should work together. I think everybody should try to do th ier own work even if they don't come up with 186

results. There is too much working together in college. With use of a text allowed, they would a ll be going to i t . That would take away the c re a tiv ity from the course. If you came up with a solu­ tio n , you would s t i l l be doing i t according to the text. I think what this course enables one to have is the fact that you learn to do for yourself. You learn to actually do the proof and go through the methods.

3. Working together probably wouldn't be too good. There were times I would have liked to have talked to somebody, but that's what we did in class. If we had a question, we asked it in class. You might get furth er though. A text might help sometimes, but it might become a crutch. Plus, on all the hard ones, you wouldn't get any v a rie ty . They would a ll have what was in the te x t. k. Working together would probably help. The students could get ideas from each other. Texts would also be a good idea in some places. Maybe the f i r s t part could be done without them, but then use them for the more difficult stuff.

5. I think there is a certain amount of working together as it is. You come to class, put a problem up, and somebody makes a comment about it. You synthesize this all together and get a result. So you are really working together. I really don't think there is that much to gain from working together outside of class. I think the students working on the problems alone and then coming together in class and discussing them is the strength of the course. Maybe for particular type things there might be a place for using a text.

6. It would be nice i f there was some kind of answer book. Then a student wouldn't have to collaberate with other students. He could come in and check with the instructor if he was having prob­ lems with i t . A text would be a ll right as long as they d id n 't just take it out of the text and not look at it on their own. There is no real way around i t unless the student wants to be true to himself.

7* Unrestricted working together would probably result in some stu­ dents not doing the work but ju s t getting the proofs from someone. If the students worked alone firs t, it might work out pretty well in some cases. A lot of times when you ta lk about something a ll you have to do is try to explain something to somebody else and all of a sudden you see these brand new insights into it. A cer­ tain amount of interaction might work out well. A text could do the same thing for an honest student. If he worked until he was frustrated before looking at the text it would be a great help. For me anyway, the best time to see something is when I'm frus­ trated. I f I'm working on something and can't get i t , then I read it and see it and put everything together it sticks. Whereas if I get frustrated on a problem as in this course, I can't look at a book or anythingl So I go to class the next day, and I'm half asleep, and a guy puts it on the board. Well I could care less. 187

You know I look at it and I say "yeah, th a t's gre at." And i t does not have the same e ffe c t on me.

8. I don't think working together would work. People would have the same proof. I f everybody does th e ir own work and presents it in class, then it is either right or it gets shot down.Maybe next time they will get credit for it.

What e ffe c t did my personality have on the course?

I think you made more of a point to just be sitting in the room and let everybody else run the course. Once in a while you gave hints or presented a problem.

What did you think about the hints?

They d id n 't help me at a ll. On one problem I got a better understanding from the h in t, but I s t i l l couldn't prove i t .

To what extent is the success or failure of this method of teaching dependent on the instructor?

1. Probably it is more dependent on the students. I t is dependent onthe instructor for the way you start it out and the attitude about i t . I would say about fo rty -fiv e per cent dependent on the instructor.

What effect does the instructor's personality have?

I think it has a lot of effect. I don't think you instilled fear into anybody, but your hints and things were motivating. I think your whole attitude was one that helped to motivate people.

Do you think any other members of this department would be suitable for this teaching method?

I don't really think any of them would be as suited to this teaching method. I don't think any of them could handle i t in the same way. I really used to be scared of Dr. Lightner when I was a sophomore, but I was never afraid of you.

2. It is somewhat dependent on the instructor forhin ts, etc.

What effect did the instructor's personality have?

I t made the course interesting. You were always questioning our work. You have to get along with the professor before you can take a course like this and get anything out of it.

Should I have given more hints?

One thing, on one problem I was working one way and you gave 188

a hint so I ju st dropped my way and began working on it accord­ ing to the hint. I don't know if my way was right or not. S till, I think as a professor you know a better or easier way to get to the proof. If you can channel us and ju s t channel us, I think we can s t i l l work through it and come up with the proof. I still think we will get as much out of it working from a hint as we would maybe if we worked way around and got it some other way. I think the number of hints given was ade­ quate. They occured whenever we got really bogged down. Once I s ta rt working on a method, I don't 1 ike to give up on i t . I just keep working on it and trying to push it through. But lots of time you have to be channeled, I think.

3- You could get an instructor that would say too much. You said rel­ atively little. You didn't really let out too many hot ideas. If you had an instructor that said too much, then the students would not bother doing i t . They would ju st figure they could go to class and the teacher would te ll them how to do i t . Whereas the way we had it if somebody didn't figure it out one day, then we all had to go and work on it for the next time. k. Not to a great extent.

5. Probably eighty percent the student. If he doesn't try or put forth an e ffo rt it is going to be a disaster from the beginning. It is necessary for the instructor to make critical comments at points where the student doesn't have the background to see his errors. Comments need to draw the student out rather than tell him.

6. To a degree he helps guide the progression of the course—the amount of problems that should be done within a certain time. Not a rig id set number, but an idea that perhaps we are spending too long in a certain set of proofs—a certain segment of the course— and maybe we should move on. Also his hints fo r proofs nob dy can get a foothold on help quite a b it. I think most courses are suc­ cessful or unsuccessful due to the instructor because of his way of shaping the course and presenting the course to the students.

7. Quite a bit depends on the instructor. First the difficulty of the problems on the sheets. Also it takes a certain amount of feel about how critical to be of a proof. Excessive criticism would probably ruin a lot of people's incentive. The feel he has for the students is particularly important. I think you had a f a ir ly wide balanced hierarchy of d iffic u lty in the problems.

8. Most of his work is in making out the sheets. Also, sometimes speeding up the class when i t should be sped up, making sure it is running smoothly, maybe putting a l i t t l e pressure on.

What did you think about the pace of the course? 189

Some days we sat there and saw three or four proofs of the same thing. Maybe we didn't need to present all the versions. Most of the days when we spent a lot of time we spent a whole class or more than a class on one problem. I think i t was worthwhile because some of the problems I was really lost on and I would still be lost on them if we hadn't taken the time.

Do you consider the course to have been a success or fa ilu re for your­ self? for the class?

1. Even though I d id n 't learn as much as I wanted to or think I should have, I think it was a success. I think it changed my attitudes toward mathematics and towards proof. I think they were good changes. I t was a success for the class too. I t taught everyone to work more independently, to rely on themselves.

2. A success. I know there were a couple of places where I would not think it was, but overall it was a success for the class.

Do you think I should continue this method with my real analysis classes in the future?

Yes. 1 think everyone should be exposed to the method. I think they would learn just as much as reading it out of a book. In fa c t, they would learn more. They would learn more about doing proofs and learning to read mathematics.

Is the method better for the good students, the poor students, the average students, or is there a difference?

I think the good students probably get more out of it. The good students will be working a lot of the proofs and will be trying to do most of the proofs, while the poor student won't know which way to go; although, the poor student w ill prob­ ably get more out of what is presented on the board by the other students. By seeing what the other students did, he w ill learn from th is . But I don't think the poor student really learns his own way; wher as, the good student maybe learns a ll kinds of s tu ff by doing i t . It is n 't necessarily just the poor student who does this way. Some students just can't work unless they see something. They read something and study i t and do good that way. A student must have crea­ t iv it y to do good under this method.

How much effect does the student's personality have?

It does have some e ffe c t. A shy person might not be able to get up in front of the class like this. 190

3. A success. The class liked it and benefited from i t . They generally thought it was not real hard. It was probably success­ ful for most of them. k. Not a success, but not really a fa ilu re e ith e r. I have learned some real analysis. Overall for the class i t was not a success. It was a success mainly for three students.

5. For myself I think it was a very good success. For some people, I think it wasn't. For the class as a whole, I think it was successful.

6. Basically a success in that it gave me experience in working with proof. For the class a success, because I feel that other students feel much the same way. They had not had much in the way of mathe­ matical proofs, especially proofs that we worked out. We learned how to w rite proofs which enables us to read mathematics better.

7. Successful. It has got me to think more on my own, be more creative. It was unsuccessful for some members of the class. I think some really flopped out. Overall it was probably a success.

8. A success.

Did you spend more or less time on this course than you usually do on an advanced mathematics course?

1. About the same on the average.

2. I think I spent as much or a little bit more time overall.

3. About the same. k. About the same o verall. The last two weeks, though, I d id n 't do much.

5. Probably about average. I spent more time on abstract algebra.

6. More.

7. It 's hard to say. The same or a l i t t l e more. It 's hard to say since my work was e rra tic . When I was whipping out problems I would go right on through a section and then wait fo r the class to catch up. Also, there was a lot of time when I would sit for a long time hung up on a problem, which is not like studying fo r a course.

8. More.

What was your primary motivation for studying?

I. Personal motivation to succeed. 191

2. I lik e the challenge of proving problems.

3. That was what we were getting graded on.

5. I wasn't bored like I am in most courses. It was interesting. I would get a good feeling of accomplishment a fte r doing a prob­ lem. I enjoyed going into class and having this interaction.

6. I liked the idea of generating a system by using proofs that I had worked on e a rlie r to prove other theorems.

7. Doing the problems. I enjoy seeing if I can get a proof and the sense of achievement that goes with it.

To what extent were you influenced by other members of the class?

1. There was an element of competition. It was not overpowering. There was a desire to not be a failure all the time.

2. I know one thing. When you do a proof, you better be sure i t is rig h t, because you knew that i f you were a l i t t l e b it o ff or something was wrong with your proof, there was a loophole or something, they would probably catch i t . I wanted to be really sure i f I presented something at the board.

3. I wouldn't say that they motivated me. Some days I got the feeling that [another student] and I were the only ones working. I would think that the people at the bottom would be motivated by the people at the top. That would be a good type of course to have two sections. Put a ll the smart ones in one section and all the slow ones in the other section. They would probably get more out of i t . The fast ones could go fa s te r. The slow ones could ask questions and not take other peoples time.

5. Probably very little.

6. I was in comparing myself with the other students less confident in putting any of my proofs up.

Were you intimidated by the other students?

Intimidated is not the word. They were not going to come out and hit me or shoot me or anything. But I felt that I was less able to prove the material correctly in comparing myself to the others.

7. [Two students] kind of motivated me. It they weren't in the class, I don't know if I would have done as much or not. 192

Do you feel that your grade accurately reflected your work and achievement in the course?

1. I think so, in relation to everybody e ls e 's .

2. Yeah. The last two weeks I d id n 't spend much time. I got bogged down a time or two.

3. Yes. I f the course had been harder, I wouldn't have been too upset. If it had gone faster and I would have had to spend more time on it, it wouldn't have bothered me too much, unless it was a whole lot harder. k. It was what I expected. I don't really relish the thought of it, but I don't think i t would be f a ir any other way.

If I had kept the same format, but given tests for a signifi­ cant portion of the grade do you think this would have helped?

Gradewise, I think it would. I don't know whether it would help the course. I think i t would make you more sure of how you were doing.

5. I feel I worked hard and contributed as much to the class as any­ one, so the grade was justified.

6. I think i t is apropo. I don't like i t , but I don't have any choice. It is not unfair or anything like that. I don't like it, because it is not helping me materially. But that is not what I took the course fo r. That is not what a grade is fo r.

You seem to have a noticeable lack of achievement, yet you say the course was successful. Isn't this paradoxical?

No, the success is in the experience that I gained through i t . Not always getting a right answer doesn't mean that you have not attained any success. It is not always the right answer that counts. I guess I sound like more of a realist than anything else.

7. Yes. I put an awful lo t of work into i t .

8. Yes.

Do you have any questions that you would like to ask me or additional comments that you would like to make?

1. I was probably more frustrated at the midterm interview. I was pretty snowed under with work the f i r s t part of the semester.

2. My diary is not very accurate. I didn't carry it with me everytime 193

I started thinking about analysis. I might be down in the g r ill and maybe I would think of something and I would w rite i t down or try i t then. Or, I would be in the lib rary and a ll of a sud­ den a thought would flash in my mind, or before class I would think of something. I had my diary in my room and I would f i l l it in when I worked at night in the room, but I wouldn't think of all the things that might have popped up during the day. h. I think I would d e fin ite ly like to try a real analysis course again, in graduate school maybe, after I get over this one. BIBLIOGRAPHY 195

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